uncf1 - Metamath Proof Explorer

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Theorem uncf1 18281

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 18279	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))
6	5	fveq2d 6910	. . 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 7448	. 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 7434	. . 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 18223	. . . 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 17908	. . . . . . . . 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 2737	. . . . . . 7 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
19	9, 17, 2, 18	1stfcl 18242	. . . . . 6 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
20	19, 4	cofucl 17933	. . . . 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 18243	. . . . 5 ⊢ (𝜑 → (𝐶 2^nd_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐷))
23	13, 14, 20, 22	prfcl 18248	. . . 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 18267	. . . 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 5724	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	12, 23, 26, 29	cofu1 17929	. . 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 2789	. 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 18245	. . . . . 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 17929	. . . . . . . 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 18237	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
36		op1stg 8026	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
37	27, 28, 36	syl2anc 584	. . . . . . . . . 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 6910	. . . . . . . 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 18240	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = (2^nd ‘⟨𝑋, 𝑌⟩))
42		op2ndg 8027	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
43	27, 28, 42	syl2anc 584	. . . . . . . 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 4881	. . . . . 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 6910	. . . 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 7434	. . . 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 2795	. . 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 18008	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51		relfunc 17907	. . . . . . 7 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52		1st2ndbr 8067	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2^nd ‘𝐺))
54	10, 50, 53	funcf1 17911	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
55	54, 27	ffvelcdmd 7105	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
56	24, 2, 3, 11, 55, 28	evlf1 18265	. . 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 395 = wceq 1540 ∈ wcel 2108 ⟨cop 4632 class class class wbr 5143 × cxp 5683 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ⟨“cs3 14881 Basecbs 17247 Hom chom 17308 Catccat 17707 Func cfunc 17899 ∘_func ccofu 17901 FuncCat cfuc 17990 ×_c cxpc 18213 1^st_F c1stf 18214 2^nd_F c2ndf 18215 ⟨,⟩_F cprf 18216 eval_F cevlf 18254 uncurry_F cuncf 18256

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-cat 17711 df-cid 17712 df-func 17903 df-cofu 17905 df-nat 17991 df-fuc 17992 df-xpc 18217 df-1stf 18218 df-2ndf 18219 df-prf 18220 df-evlf 18258 df-uncf 18260

This theorem is referenced by: curfuncf 18283 uncfcurf 18284

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