Theorem yon2 16906

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 2622	. . . . . . . . 9 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2622	. . . . . . . . 9 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 16901	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝑌) = (2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	oveqd 6667	. . . . . 6 ⊢ (𝜑 → (𝑋(2nd ‘𝑌)𝑍) = (𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍))
8	7	fveq1d 6193	. . . . 5 ⊢ (𝜑 → ((𝑋(2nd ‘𝑌)𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹))
9	8	fveq1d 6193	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10		eqid 2622	. . . . 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 16382	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2622	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6201	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7100	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssid 3624	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶)
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
20	3, 4, 14, 2, 17, 19	oppchofcl 16900	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
21	3, 11	oppcbas 16378	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
23		eqid 2622	. . . . 5 ⊢ (Id‘(oppCat‘𝐶)) = (Id‘(oppCat‘𝐶))
24		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
25		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
26		yon2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))
27		eqid 2622	. . . . 5 ⊢ ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)
28		yon12.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
29	10, 11, 2, 13, 20, 21, 22, 23, 24, 25, 26, 27, 28	curf2val 16870	. . . 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)))
30	9, 29	eqtrd 2656	. . 3 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
31	30	fveq1d 6193	. 2 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺))
32		eqid 2622	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
33		eqid 2622	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
34	22, 3	oppchom 16375	. . . 4 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
35	26, 34	syl6eleqr 2712	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
36	21, 32, 23, 13, 28	catidcl 16343	. . 3 ⊢ (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
37		yon2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))
38	22, 3	oppchom 16375	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
39	37, 38	syl6eleqr 2712	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
40	4, 13, 21, 32, 24, 28, 25, 28, 33, 35, 36, 39	hof2 16897	. 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))
41	21, 32, 23, 13, 24, 33, 28, 39	catlid 16344	. . . 4 ⊢ (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
42	41	oveq1d 6665	. . 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))
43		yon12.x	. . . 4 ⊢ · = (comp‘𝐶)
44	11, 43, 3, 25, 24, 28	oppcco 16377	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
45	42, 44	eqtrd 2656	. 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))
46	31, 40, 45	3eqtrd 2660	1 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183  ran crn 5115  ‘cfv 5888  (class class class)co 6650  2^nd c2nd 7167  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326  Hom_f chomf 16327  oppCatcoppc 16371  SetCatcsetc 16725   curry_F ccurf 16850  Hom_Fchof 16888  Yoncyon 16889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-oppc 16372  df-func 16518  df-setc 16726  df-xpc 16812  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedalem3b  16919  yonffthlem  16922