uncf1 - Metamath Proof Explorer

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Theorem uncf1 17744

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 17742	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))
6	5	fveq2d 6721	. . 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 7230	. 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 7216	. . 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 2737	. . . . 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 17685	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
13		eqid 2737	. . . . 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 2737	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c 𝐷) = ((𝐷 FuncCat 𝐸) ×_c 𝐷)
15		funcrcl 17369	. . . . . . . . 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 498	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
18		eqid 2737	. . . . . . 7 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
19	9, 17, 2, 18	1stfcl 17704	. . . . . 6 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
20	19, 4	cofucl 17394	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func (𝐶 1^st_F 𝐷)) ∈ ((𝐶 ×_c 𝐷) Func (𝐷 FuncCat 𝐸)))
21		eqid 2737	. . . . . 6 ⊢ (𝐶 2^nd_F 𝐷) = (𝐶 2^nd_F 𝐷)
22	9, 17, 2, 21	2ndfcl 17705	. . . . 5 ⊢ (𝜑 → (𝐶 2^nd_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐷))
23	13, 14, 20, 22	prfcl 17710	. . . 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 2737	. . . . 5 ⊢ (𝐷 eval_F 𝐸) = (𝐷 eval_F 𝐸)
25		eqid 2737	. . . . 5 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
26	24, 25, 2, 3	evlfcl 17730	. . . 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 5589	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	12, 23, 26, 29	cofu1 17390	. . 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	syl5eq 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 2737	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
33	13, 12, 32, 20, 22, 29	prf1 17707	. . . . . 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 17390	. . . . . . . 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 17699	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
36		op1stg 7773	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
37	27, 28, 36	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
38	35, 37	eqtrd 2777	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
39	38	fveq2d 6721	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐺)‘𝑋))
40	34, 39	eqtrd 2777	. . . . . . 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 17702	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = (2^nd ‘⟨𝑋, 𝑌⟩))
42		op2ndg 7774	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
43	27, 28, 42	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
44	41, 43	eqtrd 2777	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
45	40, 44	opeq12d 4792	. . . . . 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 2777	. . . . 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 6721	. . . 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 7216	. . . 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 2796	. . 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 17468	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51		relfunc 17368	. . . . . . 7 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52		1st2ndbr 7813	. . . . . . 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 590	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2^nd ‘𝐺))
54	10, 50, 53	funcf1 17372	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
55	54, 27	ffvelrnd 6905	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
56	24, 2, 3, 11, 55, 28	evlf1 17728	. . 3 ⊢ (𝜑 → (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))
57	49, 56	eqtrd 2777	. 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 2781	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 399 = wceq 1543 ∈ wcel 2110 ⟨cop 4547 class class class wbr 5053 × cxp 5549 Rel wrel 5556 ‘cfv 6380 (class class class)co 7213 1^st c1st 7759 2^nd c2nd 7760 ⟨“cs3 14407 Basecbs 16760 Hom chom 16813 Catccat 17167 Func cfunc 17360 ∘_func ccofu 17362 FuncCat cfuc 17449 ×_c cxpc 17675 1^st_F c1stf 17676 2^nd_F c2ndf 17677 ⟨,⟩_F cprf 17678 eval_F cevlf 17717 uncurry_F cuncf 17719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-s2 14413 df-s3 14414 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-hom 16826 df-cco 16827 df-cat 17171 df-cid 17172 df-func 17364 df-cofu 17366 df-nat 17450 df-fuc 17451 df-xpc 17679 df-1stf 17680 df-2ndf 17681 df-prf 17682 df-evlf 17721 df-uncf 17723

This theorem is referenced by: curfuncf 17746 uncfcurf 17747

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