Theorem reldmtpos 8234

Description: Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
reldmtpos	⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Proof of Theorem reldmtpos

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

Step	Hyp	Ref	Expression
1		0ex 5302	. . . . 5 ⊢ ∅ ∈ V
2	1	eldm 5898	. . . 4 ⊢ (∅ ∈ dom 𝐹 ↔ ∃𝑦∅𝐹𝑦)
3		brtpos0 8233	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
4	3	elv 3476	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5		0nelrel0 5733	. . . . . . 7 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6		vex 3474	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	1, 6	breldm 5906	. . . . . . 7 ⊢ (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
8	5, 7	nsyl3 138	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
9	4, 8	sylbir 234	. . . . 5 ⊢ (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
10	9	exlimiv 1926	. . . 4 ⊢ (∃𝑦∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
11	2, 10	sylbi 216	. . 3 ⊢ (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
12	11	con2i 139	. 2 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13		vex 3474	. . . . . 6 ⊢ 𝑥 ∈ V
14	13	eldm 5898	. . . . 5 ⊢ (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15		relcnv 6103	. . . . . . . . . . 11 ⊢ Rel ◡dom 𝐹
16		df-rel 5680	. . . . . . . . . . 11 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
17	15, 16	mpbi 229	. . . . . . . . . 10 ⊢ ◡dom 𝐹 ⊆ (V × V)
18	17	sseli 3975	. . . . . . . . 9 ⊢ (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V))
19	18	a1i 11	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V)))
20		elsni 4642	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
21	20	breq1d 5153	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
22	1, 6	breldm 5906	. . . . . . . . . . . . 13 ⊢ (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
23	22	pm2.24d 151	. . . . . . . . . . . 12 ⊢ (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
24	4, 23	sylbi 216	. . . . . . . . . . 11 ⊢ (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
25	21, 24	biimtrdi 252	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V))))
26	25	com3l 89	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
27	26	impcom 407	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28		brtpos2 8232	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦)))
29	6, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦))
30	29	simplbi 497	. . . . . . . . . 10 ⊢ (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (◡dom 𝐹 ∪ {∅}))
31		elun 4145	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↔ (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
32	30, 31	sylib 217	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
33	32	adantl 481	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
34	19, 27, 33	mpjaod 859	. . . . . . 7 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
35	34	ex 412	. . . . . 6 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
36	35	exlimdv 1929	. . . . 5 ⊢ (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
37	14, 36	biimtrid 241	. . . 4 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ (V × V)))
38	37	ssrdv 3985	. . 3 ⊢ (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39		df-rel 5680	. . 3 ⊢ (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
40	38, 39	sylibr 233	. 2 ⊢ (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
41	12, 40	impbii 208	1 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846  ∃wex 1774   ∈ wcel 2099  Vcvv 3470   ∪ cun 3943   ⊆ wss 3945  ∅c0 4319  {csn 4625  ∪ cuni 4904   class class class wbr 5143   × cxp 5671  ^◡ccnv 5672  dom cdm 5673  Rel wrel 5678  tpos ctpos 8225

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 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-tpos 8226

This theorem is referenced by:  dmtpos  8238