Theorem kgencn3 23449

Description: The set of continuous functions from 𝐽 to 𝐾 is unaffected by k-ification of 𝐾, if 𝐽 is already compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
kgencn3	⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Proof of Theorem kgencn3

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2727	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2727	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23137	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶∪ 𝐾)
4	3	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶∪ 𝐾)
5		cnvimass 6079	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
6	4	fdmd 6727	. . . . . . . . . 10 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = ∪ 𝐽)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = ∪ 𝐽)
8	5, 7	sseqtrid 4030	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽)
9		cnvresima 6228	. . . . . . . . . . . 12 ⊢ (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦)
10	4	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑓:∪ 𝐽⟶∪ 𝐾)
11		ffun 6719	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐽⟶∪ 𝐾 → Fun 𝑓)
12		inpreima 7067	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝑓 → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))))
13	10, 11, 12	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))))
14	13	ineq1d 4207	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦))
15		in32 4217	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))
16		ssrin 4229	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝑓 “ 𝑥) ⊆ dom 𝑓 → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦))
17	5, 16	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦)
18		dminss 6151	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
19	17, 18	sstri 3987	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))
20	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)))
21		df-ss 3961	. . . . . . . . . . . . . . 15 ⊢ (((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) ↔ (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
22	20, 21	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
23	15, 22	eqtrid 2779	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
24	14, 23	eqtrd 2767	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
25	9, 24	eqtrid 2779	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦))
26		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
27	26	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑓 ∈ (𝐽 Cn 𝐾))
28		elpwi 4605	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
29	28	ad2antrl 727	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑦 ⊆ ∪ 𝐽)
30	1	cnrest 23176	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ⊆ ∪ 𝐽) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾))
31	27, 29, 30	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾))
32		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
33	32	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ Top)
34		toptopon2 22807	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
35	33, 34	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
36		df-ima 5685	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
37	36	eqimss2i 4039	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦)
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦))
39		imassrn 6068	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓
40	10	frnd 6724	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ran 𝑓 ⊆ ∪ 𝐾)
41	39, 40	sstrid 3989	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 “ 𝑦) ⊆ ∪ 𝐾)
42		cnrest2 23177	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦) ∧ (𝑓 “ 𝑦) ⊆ ∪ 𝐾) → ((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦)))))
43	35, 38, 41, 42	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦)))))
44	31, 43	mpbid 231	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))))
45		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐾))
46		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝐽 ↾t 𝑦) ∈ Comp)
47		imacmp 23288	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑦) ∈ Comp) → (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp)
48	27, 46, 47	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp)
49		kgeni 23428	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐾) ∧ (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp) → (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦)))
50	45, 48, 49	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦)))
51		cnima 23156	. . . . . . . . . . . 12 ⊢ (((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))) ∧ (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦))) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦))
52	44, 50, 51	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦))
53	25, 52	eqeltrrd 2829	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽 ∧ (𝐽 ↾t 𝑦) ∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))
54	53	expr 456	. . . . . . . . 9 ⊢ (((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 ∪ 𝐽) → ((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
55	54	ralrimiva 3141	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))
56		kgentop 23433	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
57	56	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top)
58		toptopon2 22807	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
59	57, 58	sylib 217	. . . . . . . . 9 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
60		elkgen 23427	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
61	59, 60	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)))))
62	8, 55, 61	mpbir2and 712	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽))
63		kgenidm 23438	. . . . . . . 8 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
64	63	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽)
65	62, 64	eleqtrd 2830	. . . . . 6 ⊢ ((((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽)
66	65	ralrimiva 3141	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)
67	56, 58	sylib 217	. . . . . . 7 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ (TopOn‘∪ 𝐽))
68		kgentopon 23429	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
69	34, 68	sylbi 216	. . . . . . 7 ⊢ (𝐾 ∈ Top → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
70		iscn 23126	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
71	67, 69, 70	syl2an 595	. . . . . 6 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
72	71	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽)))
73	4, 66, 72	mpbir2and 712	. . . 4 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))
74	73	ex 412	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))))
75	74	ssrdv 3984	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾)))
76	69	adantl 481	. . . 4 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
77		toponcom 22817	. . . 4 ⊢ ((𝐾 ∈ Top ∧ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)))
78	32, 76, 77	syl2anc 583	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)))
79		kgenss 23434	. . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ⊆ (𝑘Gen‘𝐾))
80	79	adantl 481	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → 𝐾 ⊆ (𝑘Gen‘𝐾))
81		eqid 2727	. . . 4 ⊢ ∪ (𝑘Gen‘𝐾) = ∪ (𝑘Gen‘𝐾)
82	81	cnss2 23168	. . 3 ⊢ ((𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
83	78, 80, 82	syl2anc 583	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾))
84	75, 83	eqssd 3995	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ∩ cin 3943   ⊆ wss 3944  𝒫 cpw 4598  ∪ cuni 4903  ^◡ccnv 5671  dom cdm 5672  ran crn 5673   ↾ cres 5674   “ cima 5675  Fun wfun 6536  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ↾_t crest 17393  Topctop 22782  TopOnctopon 22799   Cn ccn 23115  Compccmp 23277  𝑘Genckgen 23424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-fin 8959  df-fi 9426  df-rest 17395  df-topgen 17416  df-top 22783  df-topon 22800  df-bases 22836  df-cn 23118  df-cmp 23278  df-kgen 23425

This theorem is referenced by:  kgen2cn  23450  txkgen  23543  qtopkgen  23601