uncf1 - Metamath Proof Explorer

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Theorem uncf1 18306

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 18304	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐷 eval_F 𝐸) ∘_func ((𝐺 ∘_func (𝐶 1^st_F 𝐷)) ⟨,⟩_F (𝐶 2^nd_F 𝐷))))
6	5	fveq2d 6924	. . 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 7465	. 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 7451	. . 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 2740	. . . . 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 18247	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
13		eqid 2740	. . . . 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 2740	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c 𝐷) = ((𝐷 FuncCat 𝐸) ×_c 𝐷)
15		funcrcl 17927	. . . . . . . . 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 2740	. . . . . . 7 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
19	9, 17, 2, 18	1stfcl 18266	. . . . . 6 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
20	19, 4	cofucl 17952	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func (𝐶 1^st_F 𝐷)) ∈ ((𝐶 ×_c 𝐷) Func (𝐷 FuncCat 𝐸)))
21		eqid 2740	. . . . . 6 ⊢ (𝐶 2^nd_F 𝐷) = (𝐶 2^nd_F 𝐷)
22	9, 17, 2, 21	2ndfcl 18267	. . . . 5 ⊢ (𝜑 → (𝐶 2^nd_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐷))
23	13, 14, 20, 22	prfcl 18272	. . . 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 2740	. . . . 5 ⊢ (𝐷 eval_F 𝐸) = (𝐷 eval_F 𝐸)
25		eqid 2740	. . . . 5 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
26	24, 25, 2, 3	evlfcl 18292	. . . 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 5739	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	12, 23, 26, 29	cofu1 17948	. . 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 2792	. 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 2740	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
33	13, 12, 32, 20, 22, 29	prf1 18269	. . . . . 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 17948	. . . . . . . 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 18261	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
36		op1stg 8042	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
39	38	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐺)‘𝑋))
40	34, 39	eqtrd 2780	. . . . . . 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 18264	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = (2^nd ‘⟨𝑋, 𝑌⟩))
42		op2ndg 8043	. . . . . . . . 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 2780	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝐶 2^nd_F 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
45	40, 44	opeq12d 4905	. . . . . 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 2780	. . . . 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 6924	. . . 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 7451	. . . 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 2798	. . 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 18029	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51		relfunc 17926	. . . . . . 7 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52		1st2ndbr 8083	. . . . . . 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 17930	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
55	54, 27	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
56	24, 2, 3, 11, 55, 28	evlf1 18290	. . 3 ⊢ (𝜑 → (((1^st ‘𝐺)‘𝑋)(1^st ‘(𝐷 eval_F 𝐸))𝑌) = ((1^st ‘((1^st ‘𝐺)‘𝑋))‘𝑌))
57	49, 56	eqtrd 2780	. 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 2784	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 1537 ∈ wcel 2108 ⟨cop 4654 class class class wbr 5166 × cxp 5698 Rel wrel 5705 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ⟨“cs3 14891 Basecbs 17258 Hom chom 17322 Catccat 17722 Func cfunc 17918 ∘_func ccofu 17920 FuncCat cfuc 18010 ×_c cxpc 18237 1^st_F c1stf 18238 2^nd_F c2ndf 18239 ⟨,⟩_F cprf 18240 eval_F cevlf 18279 uncurry_F cuncf 18281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-s2 14897 df-s3 14898 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-hom 17335 df-cco 17336 df-cat 17726 df-cid 17727 df-func 17922 df-cofu 17924 df-nat 18011 df-fuc 18012 df-xpc 18241 df-1stf 18242 df-2ndf 18243 df-prf 18244 df-evlf 18283 df-uncf 18285

This theorem is referenced by: curfuncf 18308 uncfcurf 18309

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