Theorem yon2 18445

Description: Value of the Yoneda embedding at a morphism. (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)
yon2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))
yon2.g	⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))

Assertion

Ref	Expression
yon2	⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

Proof of Theorem yon2

Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . . . 9 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2760	. . . . . . . . 9 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2760	. . . . . . . . 9 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18440	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝑌) = (2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	oveqd 7437	. . . . . 6 ⊢ (𝜑 → (𝑋(2nd ‘𝑌)𝑍) = (𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍))
8	7	fveq1d 6887	. . . . 5 ⊢ (𝜑 → ((𝑋(2nd ‘𝑌)𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹))
9	8	fveq1d 6887	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10		eqid 2760	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
12	3	oppccat 17901	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 18	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2760	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6898	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7922	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3931	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 18439	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17897	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
22		eqid 2760	. . . . 5 ⊢ (Id‘(oppCat‘𝐶)) = (Id‘(oppCat‘𝐶))
23		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		yon2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))
26		eqid 2760	. . . . 5 ⊢ ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)
27		yon12.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
28	10, 11, 2, 13, 19, 20, 21, 22, 23, 24, 25, 26, 27	curf2val 18409	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
29	9, 28	eqtrd 2795	. . 3 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
30	29	fveq1d 6887	. 2 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺))
31		eqid 2760	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
32		eqid 2760	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33	21, 3	oppchom 17894	. . . 4 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
34	25, 33	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
35	20, 31, 22, 13, 27	catidcl 17861	. . 3 ⊢ (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
36		yon2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))
37	21, 3	oppchom 17894	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
38	36, 37	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
39	4, 13, 20, 31, 23, 27, 24, 27, 32, 34, 35, 38	hof2 18436	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺) = ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹))
40	20, 31, 22, 13, 23, 32, 27, 38	catlid 17862	. . . 4 ⊢ (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
41	40	oveq1d 7435	. . 3 ⊢ (𝜑 → ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹))
42		yon12.x	. . . 4 ⊢ · = (comp‘𝐶)
43	11, 42, 3, 24, 23, 27	oppcco 17896	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
44	41, 43	eqtrd 2795	. 2 ⊢ (𝜑 → ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
45	30, 39, 44	3eqtrd 2799	1 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834  ran crn 5651  ‘cfv 6538  (class class class)co 7420  2^nd c2nd 8000  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Idccid 17844  Hom_f chomf 17845  oppCatcoppc 17890  SetCatcsetc 18255   curry_F ccurf 18389  Hom_Fchof 18427  Yoncyon 18428

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 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

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 4585  df-pr 4587  df-tp 4589  df-op 4835  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 8246  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-hom 17457  df-cco 17458  df-cat 17847  df-cid 17848  df-homf 17849  df-comf 17850  df-oppc 17891  df-func 18038  df-setc 18256  df-xpc 18351  df-curf 18393  df-hof 18429  df-yon 18430

This theorem is used by:  yonedalem3b  18458  yonffthlem  18461