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Theorem yon12 17503
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 2818 . . . . . . . . . 10 (oppCat‘𝐶) = (oppCat‘𝐶)
4 eqid 2818 . . . . . . . . . 10 (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
51, 2, 3, 4yonval 17499 . . . . . . . . 9 (𝜑𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
65fveq2d 6667 . . . . . . . 8 (𝜑 → (1st𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
76fveq1d 6665 . . . . . . 7 (𝜑 → ((1st𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
87fveq2d 6667 . . . . . 6 (𝜑 → (2nd ‘((1st𝑌)‘𝑋)) = (2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
98oveqd 7162 . . . . 5 (𝜑 → (𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊) = (𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊))
109fveq1d 6665 . . . 4 (𝜑 → ((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11 eqid 2818 . . . . 5 (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
12 yon11.b . . . . 5 𝐵 = (Base‘𝐶)
133oppccat 16980 . . . . . 6 (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
142, 13syl 17 . . . . 5 (𝜑 → (oppCat‘𝐶) ∈ Cat)
15 eqid 2818 . . . . . 6 (SetCat‘ran (Homf𝐶)) = (SetCat‘ran (Homf𝐶))
16 fvex 6676 . . . . . . . 8 (Homf𝐶) ∈ V
1716rnex 7606 . . . . . . 7 ran (Homf𝐶) ∈ V
1817a1i 11 . . . . . 6 (𝜑 → ran (Homf𝐶) ∈ V)
19 ssidd 3987 . . . . . 6 (𝜑 → ran (Homf𝐶) ⊆ ran (Homf𝐶))
203, 4, 15, 2, 18, 19oppchofcl 17498 . . . . 5 (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf𝐶))))
213, 12oppcbas 16976 . . . . 5 𝐵 = (Base‘(oppCat‘𝐶))
22 yon11.p . . . . 5 (𝜑𝑋𝐵)
23 eqid 2818 . . . . 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 2818 . . . . 5 (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26 eqid 2818 . . . . 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 16973 . . . . . 6 (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
3128, 30eleqtrrdi 2921 . . . . 5 (𝜑𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
3211, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31curf12 17465 . . . 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 2853 . . 3 (𝜑 → ((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
3433fveq1d 6665 . 2 (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺))
35 eqid 2818 . . 3 (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
3612, 29, 26, 2, 22catidcl 16941 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
3729, 3oppchom 16973 . . . 4 (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
3836, 37eleqtrrdi 2921 . . 3 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39 yon12.g . . . 4 (𝜑𝐺 ∈ (𝑍𝐻𝑋))
4029, 3oppchom 16973 . . . 4 (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
4139, 40eleqtrrdi 2921 . . 3 (𝜑𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
424, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41hof2 17495 . 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 16975 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
4544oveq1d 7160 . . 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 16975 . . 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 16944 . . . 4 (𝜑 → (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
4812, 29, 26, 2, 27, 43, 22, 47catlid 16942 . . 3 (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋· 𝑋)(𝐺(⟨𝑊, 𝑍· 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))
4945, 46, 483eqtrd 2857 . 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 2857 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 1528  wcel 2105  Vcvv 3492  cop 4563  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924  Homf chomf 16925  oppCatcoppc 16969  SetCatcsetc 17323   curryF ccurf 17448  HomFchof 17486  Yoncyon 17487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-homf 16929  df-comf 16930  df-oppc 16970  df-func 17116  df-setc 17324  df-xpc 17410  df-curf 17452  df-hof 17488  df-yon 17489
This theorem is referenced by:  yonedalem4c  17515  yonedalem3b  17517  yonedainv  17519  yonffthlem  17520
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