yon12 - Metamath Proof Explorer

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Theorem yon12 18214

Description: Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)

**Hypotheses**
Ref	Expression
yon11.y	⊢ 𝑌 = (Yon‘𝐶)
yon11.b	⊢ 𝐵 = (Base‘𝐶)
yon11.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yon11.p	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yon11.h	⊢ 𝐻 = (Hom ‘𝐶)
yon11.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
yon12.x	⊢ · = (comp‘𝐶)
yon12.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
yon12.f	⊢ (𝜑 → 𝐹 ∈ (𝑊𝐻𝑍))
yon12.g	⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))

**Assertion**
Ref	Expression
yon12	⊢ (𝜑 → (((𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

**Proof of Theorem yon12**
Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2732	. . . . . . . . . 10 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2732	. . . . . . . . . 10 ⊢ (Hom_F‘(oppCat‘𝐶)) = (Hom_F‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18210	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))
6	5	fveq2d 6892	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌) = (1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶)))))
7	6	fveq1d 6890	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) = ((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (2^nd ‘((1^st ‘𝑌)‘𝑋)) = (2^nd ‘((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋)))
9	8	oveqd 7422	. . . . 5 ⊢ (𝜑 → (𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊) = (𝑍(2^nd ‘((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋))𝑊))
10	9	fveq1d 6890	. . . 4 ⊢ (𝜑 → ((𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2^nd ‘((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11		eqid 2732	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶)))
12		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
13	3	oppccat 17664	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
14	2, 13	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
15		eqid 2732	. . . . . 6 ⊢ (SetCat‘ran (Hom_f ‘𝐶)) = (SetCat‘ran (Hom_f ‘𝐶))
16		fvex 6901	. . . . . . . 8 ⊢ (Hom_f ‘𝐶) ∈ V
17	16	rnex 7899	. . . . . . 7 ⊢ ran (Hom_f ‘𝐶) ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ∈ V)
19		ssidd 4004	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ ran (Hom_f ‘𝐶))
20	3, 4, 15, 2, 18, 19	oppchofcl 18209	. . . . 5 ⊢ (𝜑 → (Hom_F‘(oppCat‘𝐶)) ∈ ((𝐶 ×_c (oppCat‘𝐶)) Func (SetCat‘ran (Hom_f ‘𝐶))))
21	3, 12	oppcbas 17659	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2732	. . . . 5 ⊢ ((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋) = ((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋)
24		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		eqid 2732	. . . . 5 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26		eqid 2732	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		yon12.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
28		yon12.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑊𝐻𝑍))
29		yon11.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
30	29, 3	oppchom 17656	. . . . . 6 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
31	28, 30	eleqtrrdi 2844	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
32	11, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31	curf12 18176	. . . 4 ⊢ (𝜑 → ((𝑍(2^nd ‘((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
33	10, 32	eqtrd 2772	. . 3 ⊢ (𝜑 → ((𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
34	33	fveq1d 6890	. 2 ⊢ (𝜑 → (((𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺))
35		eqid 2732	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
36	12, 29, 26, 2, 22	catidcl 17622	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
37	29, 3	oppchom 17656	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
38	36, 37	eleqtrrdi 2844	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39		yon12.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
40	29, 3	oppchom 17656	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
41	39, 40	eleqtrrdi 2844	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
42	4, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41	hof2 18206	. 2 ⊢ (𝜑 → ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺) = ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)))
43		yon12.x	. . . . 5 ⊢ · = (comp‘𝐶)
44	12, 43, 3, 22, 24, 27	oppcco 17658	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
45	44	oveq1d 7420	. . 3 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = ((𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)))
46	12, 43, 3, 22, 22, 27	oppcco 17658	. . 3 ⊢ (𝜑 → ((𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)))
47	12, 29, 43, 2, 27, 24, 22, 28, 39	catcocl 17625	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
48	12, 29, 26, 2, 27, 43, 22, 47	catlid 17623	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
49	45, 46, 48	3eqtrd 2776	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
50	34, 42, 49	3eqtrd 2776	1 ⊢ (𝜑 → (((𝑍(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 ran crn 5676 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17140 Hom chom 17204 compcco 17205 Catccat 17604 Idccid 17605 Hom_f chomf 17606 oppCatcoppc 17651 SetCatcsetc 18021 curry_F ccurf 18159 Hom_Fchof 18197 Yoncyon 18198

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-cat 17608 df-cid 17609 df-homf 17610 df-comf 17611 df-oppc 17652 df-func 17804 df-setc 18022 df-xpc 18120 df-curf 18163 df-hof 18199 df-yon 18200

This theorem is referenced by: yonedalem4c 18226 yonedalem3b 18228 yonedainv 18230 yonffthlem 18231

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