uncf1 - Metamath Proof Explorer

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Theorem uncf1 18222

Description: Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)

**Hypotheses**
Ref	Expression
uncfval.g	⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurry_F 𝐺)
uncfval.c	⊢ (𝜑 → 𝐷 ∈ Cat)
uncfval.d	⊢ (𝜑 → 𝐸 ∈ Cat)
uncfval.f	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
uncf1.a	⊢ 𝐴 = (Base‘𝐶)
uncf1.b	⊢ 𝐵 = (Base‘𝐷)
uncf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
uncf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
uncf1	⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))

**Proof of Theorem uncf1**
Step	Hyp	Ref	Expression
1		uncfval.g	. . . . 5 ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurry_F 𝐺)
2		uncfval.c	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
3		uncfval.d	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
4		uncfval.f	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
5	1, 2, 3, 4	uncfval 18220	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))
6	5	fveq2d 6896	. . 3 ⊢ (𝜑 → (1^st ‘𝐹) = (1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))))
7	6	oveqd 7432	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) = (𝑋(1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))𝑌))
8		df-ov 7418	. . 3 ⊢ (𝑋(1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))𝑌) = ((1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))‘⟨𝑋, 𝑌⟩)
9		eqid 2728	. . . . 5 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
10		uncf1.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
11		uncf1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
12	9, 10, 11	xpcbas 18163	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
13		eqid 2728	. . . . 5 ⊢ ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)) = ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))
14		eqid 2728	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c 𝐷) = ((𝐷 FuncCat 𝐸) ×_c 𝐷)
15		funcrcl 17843	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
16	4, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
17	16	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
18		eqid 2728	. . . . . . 7 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
19	9, 17, 2, 18	1stfcl 18182	. . . . . 6 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
20	19, 4	cofucl 17868	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func (𝐶 1^st_F 𝐷)) ∈ ((𝐶 ×_c 𝐷) Func (𝐷 FuncCat 𝐸)))
21		eqid 2728	. . . . . 6 ⊢ (𝐶 2^nd_F 𝐷) = (𝐶 2^nd_F 𝐷)
22	9, 17, 2, 21	2ndfcl 18183	. . . . 5 ⊢ (𝜑 → (𝐶 2^nd_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐷))
23	13, 14, 20, 22	prfcl 18188	. . . 4 ⊢ (𝜑 → ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)) ∈ ((𝐶 ×_c 𝐷) Func ((𝐷 FuncCat 𝐸) ×_c 𝐷)))
24		eqid 2728	. . . . 5 ⊢ (𝐷 eval_F 𝐸) = (𝐷 eval_F 𝐸)
25		eqid 2728	. . . . 5 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
26	24, 25, 2, 3	evlfcl 18208	. . . 4 ⊢ (𝜑 → (𝐷 eval_F 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×_c 𝐷) Func 𝐸))
27		uncf1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
28		uncf1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
29	27, 28	opelxpd 5712	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	12, 23, 26, 29	cofu1 17864	. . 3 ⊢ (𝜑 → ((1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))‘⟨𝑋, 𝑌⟩) = ((1^st ‘(𝐷 eval_F 𝐸))‘((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩)))
31	8, 30	eqtrid 2780	. 2 ⊢ (𝜑 → (𝑋(1^st ‘((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))𝑌) = ((1^st ‘(𝐷 eval_F 𝐸))‘((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩)))
32		eqid 2728	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
33	13, 12, 32, 20, 22, 29	prf1 18185	. . . . . 6 ⊢ (𝜑 → ((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1^st ‘(𝐺 ∘_func (𝐶 1^st_F 𝐷)))‘⟨𝑋, 𝑌⟩), ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩)⟩)
34	12, 19, 4, 29	cofu1 17864	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func (𝐶 1^st_F 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1^st ‘𝐺)‘((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)))
35	9, 12, 32, 17, 2, 18, 29	1stf1 18177	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
36		op1stg 8000	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
37	27, 28, 36	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
38	35, 37	eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
39	38	fveq2d 6896	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐺)‘𝑋))
40	34, 39	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func (𝐶 1^st_F 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1^st ‘𝐺)‘𝑋))
41	9, 12, 32, 17, 2, 21, 29	2ndf1 18180	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = (2^nd ‘⟨𝑋, 𝑌⟩))
42		op2ndg 8001	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
43	27, 28, 42	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
44	41, 43	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
45	40, 44	opeq12d 4878	. . . . . 6 ⊢ (𝜑 → ⟨((1^st ‘(𝐺 ∘_func (𝐶 1^st_F 𝐷)))‘⟨𝑋, 𝑌⟩), ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1^st ‘𝐺)‘𝑋), 𝑌⟩)
46	33, 45	eqtrd 2768	. . . . 5 ⊢ (𝜑 → ((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1^st ‘𝐺)‘𝑋), 𝑌⟩)
47	46	fveq2d 6896	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐷 eval_F 𝐸))‘((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘(𝐷 eval_F 𝐸))‘⟨((1^st ‘𝐺)‘𝑋), 𝑌⟩))
48		df-ov 7418	. . . 4 ⊢ (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌) = ((1^st ‘(𝐷 eval_F 𝐸))‘⟨((1^st ‘𝐺)‘𝑋), 𝑌⟩)
49	47, 48	eqtr4di 2786	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐷 eval_F 𝐸))‘((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩)) = (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌))
50	25	fucbas 17945	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51		relfunc 17842	. . . . . . 7 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52		1st2ndbr 8041	. . . . . . 7 ⊢ ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1^st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2^nd ‘𝐺))
53	51, 4, 52	sylancr 586	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2^nd ‘𝐺))
54	10, 50, 53	funcf1 17846	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
55	54, 27	ffvelcdmd 7090	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
56	24, 2, 3, 11, 55, 28	evlf1 18206	. . 3 ⊢ (𝜑 → (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))
57	49, 56	eqtrd 2768	. 2 ⊢ (𝜑 → ((1^st ‘(𝐷 eval_F 𝐸))‘((1^st ‘((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))
58	7, 31, 57	3eqtrd 2772	1 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⟨cop 4631 class class class wbr 5143 × cxp 5671 Rel wrel 5678 ‘cfv 6543 (class class class)co 7415 1^st c1st 7986 2^nd c2nd 7987 ⟨“cs3 14820 Basecbs 17174 Hom chom 17238 Catccat 17638 Func cfunc 17834 ∘_func ccofu 17836 FuncCat cfuc 17926 ×_c cxpc 18153 1^st_F c1stf 18154 2^nd_F c2ndf 18155 ⟨,⟩_F cprf 18156 eval_F cevlf 18195 uncurry_F cuncf 18197

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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-s2 14826 df-s3 14827 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-hom 17251 df-cco 17252 df-cat 17642 df-cid 17643 df-func 17838 df-cofu 17840 df-nat 17927 df-fuc 17928 df-xpc 18157 df-1stf 18158 df-2ndf 18159 df-prf 18160 df-evlf 18199 df-uncf 18201

This theorem is referenced by: curfuncf 18224 uncfcurf 18225

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