Theorem yon2 17900

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 2738	. . . . . . . . 9 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2738	. . . . . . . . 9 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 17895	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6760	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝑌) = (2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	oveqd 7272	. . . . . 6 ⊢ (𝜑 → (𝑋(2nd ‘𝑌)𝑍) = (𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍))
8	7	fveq1d 6758	. . . . 5 ⊢ (𝜑 → ((𝑋(2nd ‘𝑌)𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹))
9	8	fveq1d 6758	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10		eqid 2738	. . . . 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 17350	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2738	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6769	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7733	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3940	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 17894	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17345	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
22		eqid 2738	. . . . 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 2738	. . . . 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 17864	. . . 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 2778	. . 3 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
30	29	fveq1d 6758	. 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 2738	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
32		eqid 2738	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33	21, 3	oppchom 17342	. . . 4 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
34	25, 33	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
35	20, 31, 22, 13, 27	catidcl 17308	. . 3 ⊢ (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
36		yon2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))
37	21, 3	oppchom 17342	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
38	36, 37	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
39	4, 13, 20, 31, 23, 27, 24, 27, 32, 34, 35, 38	hof2 17891	. 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 17309	. . . 4 ⊢ (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
41	40	oveq1d 7270	. . 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 17344	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
44	41, 43	eqtrd 2778	. 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 2782	1 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564  ran crn 5581  ‘cfv 6418  (class class class)co 7255  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Hom_f chomf 17292  oppCatcoppc 17337  SetCatcsetc 17706   curry_F ccurf 17844  Hom_Fchof 17882  Yoncyon 17883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-hom 16912  df-cco 16913  df-cat 17294  df-cid 17295  df-homf 17296  df-comf 17297  df-oppc 17338  df-func 17489  df-setc 17707  df-xpc 17805  df-curf 17848  df-hof 17884  df-yon 17885

This theorem is referenced by:  yonedalem3b  17913  yonffthlem  17916