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Theorem yon2 18223
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 (𝜑𝐶 ∈ Cat)
yon11.p (𝜑𝑋𝐵)
yon11.h 𝐻 = (Hom ‘𝐶)
yon11.z (𝜑𝑍𝐵)
yon12.x · = (comp‘𝐶)
yon12.w (𝜑𝑊𝐵)
yon2.f (𝜑𝐹 ∈ (𝑋𝐻𝑍))
yon2.g (𝜑𝐺 ∈ (𝑊𝐻𝑋))
Assertion
Ref Expression
yon2 (𝜑 → ((((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋· 𝑍)𝐺))

Proof of Theorem yon2
StepHypRef Expression
1 yon11.y . . . . . . . . 9 𝑌 = (Yon‘𝐶)
2 yon11.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
3 eqid 2724 . . . . . . . . 9 (oppCat‘𝐶) = (oppCat‘𝐶)
4 eqid 2724 . . . . . . . . 9 (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
51, 2, 3, 4yonval 18218 . . . . . . . 8 (𝜑𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
65fveq2d 6886 . . . . . . 7 (𝜑 → (2nd𝑌) = (2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
76oveqd 7419 . . . . . 6 (𝜑 → (𝑋(2nd𝑌)𝑍) = (𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍))
87fveq1d 6884 . . . . 5 (𝜑 → ((𝑋(2nd𝑌)𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹))
98fveq1d 6884 . . . 4 (𝜑 → (((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10 eqid 2724 . . . . 5 (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11 yon11.b . . . . 5 𝐵 = (Base‘𝐶)
123oppccat 17669 . . . . . 6 (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
132, 12syl 17 . . . . 5 (𝜑 → (oppCat‘𝐶) ∈ Cat)
14 eqid 2724 . . . . . 6 (SetCat‘ran (Homf𝐶)) = (SetCat‘ran (Homf𝐶))
15 fvex 6895 . . . . . . . 8 (Homf𝐶) ∈ V
1615rnex 7897 . . . . . . 7 ran (Homf𝐶) ∈ V
1716a1i 11 . . . . . 6 (𝜑 → ran (Homf𝐶) ∈ V)
18 ssidd 3998 . . . . . 6 (𝜑 → ran (Homf𝐶) ⊆ ran (Homf𝐶))
193, 4, 14, 2, 17, 18oppchofcl 18217 . . . . 5 (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf𝐶))))
203, 11oppcbas 17664 . . . . 5 𝐵 = (Base‘(oppCat‘𝐶))
21 yon11.h . . . . 5 𝐻 = (Hom ‘𝐶)
22 eqid 2724 . . . . 5 (Id‘(oppCat‘𝐶)) = (Id‘(oppCat‘𝐶))
23 yon11.p . . . . 5 (𝜑𝑋𝐵)
24 yon11.z . . . . 5 (𝜑𝑍𝐵)
25 yon2.f . . . . 5 (𝜑𝐹 ∈ (𝑋𝐻𝑍))
26 eqid 2724 . . . . 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 (𝜑𝑊𝐵)
2810, 11, 2, 13, 19, 20, 21, 22, 23, 24, 25, 26, 27curf2val 18187 . . . 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)))
299, 28eqtrd 2764 . . 3 (𝜑 → (((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
3029fveq1d 6884 . 2 (𝜑 → ((((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺))
31 eqid 2724 . . 3 (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
32 eqid 2724 . . 3 (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
3321, 3oppchom 17661 . . . 4 (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
3425, 33eleqtrrdi 2836 . . 3 (𝜑𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
3520, 31, 22, 13, 27catidcl 17627 . . 3 (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
36 yon2.g . . . 4 (𝜑𝐺 ∈ (𝑊𝐻𝑋))
3721, 3oppchom 17661 . . . 4 (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
3836, 37eleqtrrdi 2836 . . 3 (𝜑𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
394, 13, 20, 31, 23, 27, 24, 27, 32, 34, 35, 38hof2 18214 . 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))
4020, 31, 22, 13, 23, 32, 27, 38catlid 17628 . . . 4 (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
4140oveq1d 7417 . . 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)
4311, 42, 3, 24, 23, 27oppcco 17663 . . 3 (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋· 𝑍)𝐺))
4441, 43eqtrd 2764 . 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))
4530, 39, 443eqtrd 2768 1 (𝜑 → ((((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋· 𝑍)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3466  cop 4627  ran crn 5668  cfv 6534  (class class class)co 7402  2nd c2nd 7968  Basecbs 17145  Hom chom 17209  compcco 17210  Catccat 17609  Idccid 17610  Homf chomf 17611  oppCatcoppc 17656  SetCatcsetc 18029   curryF ccurf 18167  HomFchof 18205  Yoncyon 18206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12471  df-z 12557  df-dec 12676  df-uz 12821  df-fz 13483  df-struct 17081  df-sets 17098  df-slot 17116  df-ndx 17128  df-base 17146  df-hom 17222  df-cco 17223  df-cat 17613  df-cid 17614  df-homf 17615  df-comf 17616  df-oppc 17657  df-func 17809  df-setc 18030  df-xpc 18128  df-curf 18171  df-hof 18207  df-yon 18208
This theorem is referenced by:  yonedalem3b  18236  yonffthlem  18239
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