Theorem yon12 18317

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(2nd ‘((1st ‘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 2769	. . . . . . . . . 10 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2769	. . . . . . . . . 10 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18313	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6883	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6881	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6883	. . . . . 6 ⊢ (𝜑 → (2nd ‘((1st ‘𝑌)‘𝑋)) = (2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	oveqd 7425	. . . . 5 ⊢ (𝜑 → (𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊) = (𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊))
10	9	fveq1d 6881	. . . 4 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11		eqid 2769	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
12		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
13	3	oppccat 17774	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
14	2, 13	syl 18	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
15		eqid 2769	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
16		fvex 6892	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
17	16	rnex 7903	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
19		ssidd 3968	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
20	3, 4, 15, 2, 18, 19	oppchofcl 18312	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
21	3, 12	oppcbas 17770	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2769	. . . . 5 ⊢ ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
24		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		eqid 2769	. . . . 5 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26		eqid 2769	. . . . 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 17767	. . . . . 6 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
31	28, 30	eleqtrrdi 2880	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
32	11, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31	curf12 18279	. . . 4 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
33	10, 32	eqtrd 2804	. . 3 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
34	33	fveq1d 6881	. 2 ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺))
35		eqid 2769	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
36	12, 29, 26, 2, 22	catidcl 17734	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
37	29, 3	oppchom 17767	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
38	36, 37	eleqtrrdi 2880	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39		yon12.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
40	29, 3	oppchom 17767	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
41	39, 40	eleqtrrdi 2880	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
42	4, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41	hof2 18309	. 2 ⊢ (𝜑 → ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(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 17769	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
45	44	oveq1d 7423	. . 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 17769	. . 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 17737	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
48	12, 29, 26, 2, 27, 43, 22, 47	catlid 17735	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
49	45, 46, 48	3eqtrd 2808	. 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 2808	1 ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597  ran crn 5660  ‘cfv 6533  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17265  Hom chom 17317  compcco 17318  Catccat 17716  Idccid 17717  Hom_f chomf 17718  oppCatcoppc 17763  SetCatcsetc 18128   curry_F ccurf 18262  Hom_Fchof 18300  Yoncyon 18301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-hom 17330  df-cco 17331  df-cat 17720  df-cid 17721  df-homf 17722  df-comf 17723  df-oppc 17764  df-func 17911  df-setc 18129  df-xpc 18224  df-curf 18266  df-hof 18302  df-yon 18303

This theorem is referenced by:  yonedalem4c  18329  yonedalem3b  18331  yonedainv  18333  yonffthlem  18334