uncf1 - Metamath Proof Explorer

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Theorem uncf1 17870

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 17868	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))
6	5	fveq2d 6760	. . 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 7272	. 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 7258	. . 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 2738	. . . . 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 17811	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
13		eqid 2738	. . . . 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 2738	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c 𝐷) = ((𝐷 FuncCat 𝐸) ×_c 𝐷)
15		funcrcl 17494	. . . . . . . . 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 2738	. . . . . . 7 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
19	9, 17, 2, 18	1stfcl 17830	. . . . . 6 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
20	19, 4	cofucl 17519	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func (𝐶 1^st_F 𝐷)) ∈ ((𝐶 ×_c 𝐷) Func (𝐷 FuncCat 𝐸)))
21		eqid 2738	. . . . . 6 ⊢ (𝐶 2^nd_F 𝐷) = (𝐶 2^nd_F 𝐷)
22	9, 17, 2, 21	2ndfcl 17831	. . . . 5 ⊢ (𝜑 → (𝐶 2^nd_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐷))
23	13, 14, 20, 22	prfcl 17836	. . . 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 2738	. . . . 5 ⊢ (𝐷 eval_F 𝐸) = (𝐷 eval_F 𝐸)
25		eqid 2738	. . . . 5 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
26	24, 25, 2, 3	evlfcl 17856	. . . 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 5618	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	12, 23, 26, 29	cofu1 17515	. . 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 2790	. 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 2738	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
33	13, 12, 32, 20, 22, 29	prf1 17833	. . . . . 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 17515	. . . . . . . 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 17825	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
36		op1stg 7816	. . . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
39	38	fveq2d 6760	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐺)‘𝑋))
40	34, 39	eqtrd 2778	. . . . . . 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 17828	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = (2^nd ‘⟨𝑋, 𝑌⟩))
42		op2ndg 7817	. . . . . . . . 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 2778	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
45	40, 44	opeq12d 4809	. . . . . 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 2778	. . . . 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 6760	. . . 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 7258	. . . 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 2797	. . 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 17593	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51		relfunc 17493	. . . . . . 7 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52		1st2ndbr 7856	. . . . . . 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 17497	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
55	54, 27	ffvelrnd 6944	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
56	24, 2, 3, 11, 55, 28	evlf1 17854	. . 3 ⊢ (𝜑 → (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))
57	49, 56	eqtrd 2778	. 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 2782	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 1539 ∈ wcel 2108 ⟨cop 4564 class class class wbr 5070 × cxp 5578 Rel wrel 5585 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 ⟨“cs3 14483 Basecbs 16840 Hom chom 16899 Catccat 17290 Func cfunc 17485 ∘_func ccofu 17487 FuncCat cfuc 17574 ×_c cxpc 17801 1^st_F c1stf 17802 2^nd_F c2ndf 17803 ⟨,⟩_F cprf 17804 eval_F cevlf 17843 uncurry_F cuncf 17845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-s2 14489 df-s3 14490 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-hom 16912 df-cco 16913 df-cat 17294 df-cid 17295 df-func 17489 df-cofu 17491 df-nat 17575 df-fuc 17576 df-xpc 17805 df-1stf 17806 df-2ndf 17807 df-prf 17808 df-evlf 17847 df-uncf 17849

This theorem is referenced by: curfuncf 17872 uncfcurf 17873

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