Theorem yon12 18439

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 2760	. . . . . . . . . 10 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2760	. . . . . . . . . 10 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18435	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6889	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6887	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6889	. . . . . 6 ⊢ (𝜑 → (2nd ‘((1st ‘𝑌)‘𝑋)) = (2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	oveqd 7437	. . . . 5 ⊢ (𝜑 → (𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊) = (𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊))
10	9	fveq1d 6887	. . . 4 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11		eqid 2760	. . . . 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 17896	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
14	2, 13	syl 18	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
15		eqid 2760	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
16		fvex 6898	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
17	16	rnex 7922	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
19		ssidd 3931	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
20	3, 4, 15, 2, 18, 19	oppchofcl 18434	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
21	3, 12	oppcbas 17892	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2760	. . . . 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 2760	. . . . 5 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26		eqid 2760	. . . . 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 17889	. . . . . 6 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
31	28, 30	eleqtrrdi 2871	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
32	11, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31	curf12 18401	. . . 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 2795	. . 3 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
34	33	fveq1d 6887	. 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 2760	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
36	12, 29, 26, 2, 22	catidcl 17856	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
37	29, 3	oppchom 17889	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
38	36, 37	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39		yon12.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
40	29, 3	oppchom 17889	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
41	39, 40	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
42	4, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41	hof2 18431	. 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 17891	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
45	44	oveq1d 7435	. . 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 17891	. . 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 17859	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
48	12, 29, 26, 2, 27, 43, 22, 47	catlid 17857	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
49	45, 46, 48	3eqtrd 2799	. 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 2799	1 ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4589  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839  Hom_f chomf 17840  oppCatcoppc 17885  SetCatcsetc 18250   curry_F ccurf 18384  Hom_Fchof 18422  Yoncyon 18423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-homf 17844  df-comf 17845  df-oppc 17886  df-func 18033  df-setc 18251  df-xpc 18346  df-curf 18388  df-hof 18424  df-yon 18425

This theorem is used by:  yonedalem4c  18451  yonedalem3b  18453  yonedainv  18455  yonffthlem  18456