yon12 - Metamath Proof Explorer

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Theorem yon12 18226

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 2724	. . . . . . . . . 10 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2724	. . . . . . . . . 10 ⊢ (Hom_F‘(oppCat‘𝐶)) = (Hom_F‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18222	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))
6	5	fveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌) = (1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶)))))
7	6	fveq1d 6884	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) = ((1^st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6886	. . . . . 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 7419	. . . . 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 6884	. . . 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 2724	. . . . 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 17673	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
14	2, 13	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
15		eqid 2724	. . . . . 6 ⊢ (SetCat‘ran (Hom_f ‘𝐶)) = (SetCat‘ran (Hom_f ‘𝐶))
16		fvex 6895	. . . . . . . 8 ⊢ (Hom_f ‘𝐶) ∈ V
17	16	rnex 7897	. . . . . . 7 ⊢ ran (Hom_f ‘𝐶) ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ∈ V)
19		ssidd 3998	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ ran (Hom_f ‘𝐶))
20	3, 4, 15, 2, 18, 19	oppchofcl 18221	. . . . 5 ⊢ (𝜑 → (Hom_F‘(oppCat‘𝐶)) ∈ ((𝐶 ×_c (oppCat‘𝐶)) Func (SetCat‘ran (Hom_f ‘𝐶))))
21	3, 12	oppcbas 17668	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2724	. . . . 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 2724	. . . . 5 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26		eqid 2724	. . . . 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 17665	. . . . . 6 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
31	28, 30	eleqtrrdi 2836	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
32	11, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31	curf12 18188	. . . 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 2764	. . 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 6884	. 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 2724	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
36	12, 29, 26, 2, 22	catidcl 17631	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
37	29, 3	oppchom 17665	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
38	36, 37	eleqtrrdi 2836	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39		yon12.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
40	29, 3	oppchom 17665	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
41	39, 40	eleqtrrdi 2836	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
42	4, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41	hof2 18218	. 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 17667	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
45	44	oveq1d 7417	. . 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 17667	. . 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 17634	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
48	12, 29, 26, 2, 27, 43, 22, 47	catlid 17632	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
49	45, 46, 48	3eqtrd 2768	. 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 2768	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 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4627 ran crn 5668 ‘cfv 6534 (class class class)co 7402 1^st c1st 7967 2^nd c2nd 7968 Basecbs 17149 Hom chom 17213 compcco 17214 Catccat 17613 Idccid 17614 Hom_f chomf 17615 oppCatcoppc 17660 SetCatcsetc 18033 curry_F ccurf 18171 Hom_Fchof 18209 Yoncyon 18210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-homf 17619 df-comf 17620 df-oppc 17661 df-func 17813 df-setc 18034 df-xpc 18132 df-curf 18175 df-hof 18211 df-yon 18212

This theorem is referenced by: yonedalem4c 18238 yonedalem3b 18240 yonedainv 18242 yonffthlem 18243

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