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Theorem yon12 17581
 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 (𝜑𝐶 ∈ Cat)
yon11.p (𝜑𝑋𝐵)
yon11.h 𝐻 = (Hom ‘𝐶)
yon11.z (𝜑𝑍𝐵)
yon12.x · = (comp‘𝐶)
yon12.w (𝜑𝑊𝐵)
yon12.f (𝜑𝐹 ∈ (𝑊𝐻𝑍))
yon12.g (𝜑𝐺 ∈ (𝑍𝐻𝑋))
Assertion
Ref Expression
yon12 (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))

Proof of Theorem yon12
StepHypRef Expression
1 yon11.y . . . . . . . . . 10 𝑌 = (Yon‘𝐶)
2 yon11.c . . . . . . . . . 10 (𝜑𝐶 ∈ Cat)
3 eqid 2758 . . . . . . . . . 10 (oppCat‘𝐶) = (oppCat‘𝐶)
4 eqid 2758 . . . . . . . . . 10 (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
51, 2, 3, 4yonval 17577 . . . . . . . . 9 (𝜑𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
65fveq2d 6662 . . . . . . . 8 (𝜑 → (1st𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
76fveq1d 6660 . . . . . . 7 (𝜑 → ((1st𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
87fveq2d 6662 . . . . . 6 (𝜑 → (2nd ‘((1st𝑌)‘𝑋)) = (2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
98oveqd 7167 . . . . 5 (𝜑 → (𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊) = (𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊))
109fveq1d 6660 . . . 4 (𝜑 → ((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11 eqid 2758 . . . . 5 (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
12 yon11.b . . . . 5 𝐵 = (Base‘𝐶)
133oppccat 17050 . . . . . 6 (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
142, 13syl 17 . . . . 5 (𝜑 → (oppCat‘𝐶) ∈ Cat)
15 eqid 2758 . . . . . 6 (SetCat‘ran (Homf𝐶)) = (SetCat‘ran (Homf𝐶))
16 fvex 6671 . . . . . . . 8 (Homf𝐶) ∈ V
1716rnex 7622 . . . . . . 7 ran (Homf𝐶) ∈ V
1817a1i 11 . . . . . 6 (𝜑 → ran (Homf𝐶) ∈ V)
19 ssidd 3915 . . . . . 6 (𝜑 → ran (Homf𝐶) ⊆ ran (Homf𝐶))
203, 4, 15, 2, 18, 19oppchofcl 17576 . . . . 5 (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf𝐶))))
213, 12oppcbas 17046 . . . . 5 𝐵 = (Base‘(oppCat‘𝐶))
22 yon11.p . . . . 5 (𝜑𝑋𝐵)
23 eqid 2758 . . . . 5 ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
24 yon11.z . . . . 5 (𝜑𝑍𝐵)
25 eqid 2758 . . . . 5 (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26 eqid 2758 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
27 yon12.w . . . . 5 (𝜑𝑊𝐵)
28 yon12.f . . . . . 6 (𝜑𝐹 ∈ (𝑊𝐻𝑍))
29 yon11.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
3029, 3oppchom 17043 . . . . . 6 (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
3128, 30eleqtrrdi 2863 . . . . 5 (𝜑𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
3211, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31curf12 17543 . . . 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))
3310, 32eqtrd 2793 . . 3 (𝜑 → ((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
3433fveq1d 6660 . 2 (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺))
35 eqid 2758 . . 3 (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
3612, 29, 26, 2, 22catidcl 17011 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
3729, 3oppchom 17043 . . . 4 (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
3836, 37eleqtrrdi 2863 . . 3 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39 yon12.g . . . 4 (𝜑𝐺 ∈ (𝑍𝐻𝑋))
4029, 3oppchom 17043 . . . 4 (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
4139, 40eleqtrrdi 2863 . . 3 (𝜑𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
424, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41hof2 17573 . 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)
4412, 43, 3, 22, 24, 27oppcco 17045 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
4544oveq1d 7165 . . 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)))
4612, 43, 3, 22, 22, 27oppcco 17045 . . 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)))
4712, 29, 43, 2, 27, 24, 22, 28, 39catcocl 17014 . . . 4 (𝜑 → (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
4812, 29, 26, 2, 27, 43, 22, 47catlid 17012 . . 3 (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋· 𝑋)(𝐺(⟨𝑊, 𝑍· 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
4945, 46, 483eqtrd 2797 . 2 (𝜑 → ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
5034, 42, 493eqtrd 2797 1 (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528  ran crn 5525  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692  Basecbs 16541  Hom chom 16634  compcco 16635  Catccat 16993  Idccid 16994  Homf chomf 16995  oppCatcoppc 17039  SetCatcsetc 17401   curryF ccurf 17526  HomFchof 17564  Yoncyon 17565 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-tpos 7902  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-hom 16647  df-cco 16648  df-cat 16997  df-cid 16998  df-homf 16999  df-comf 17000  df-oppc 17040  df-func 17187  df-setc 17402  df-xpc 17488  df-curf 17530  df-hof 17566  df-yon 17567 This theorem is referenced by:  yonedalem4c  17593  yonedalem3b  17595  yonedainv  17597  yonffthlem  17598
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