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Theorem yonval 17769
Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yonval.y 𝑌 = (Yon‘𝐶)
yonval.c (𝜑𝐶 ∈ Cat)
yonval.o 𝑂 = (oppCat‘𝐶)
yonval.m 𝑀 = (HomF𝑂)
Assertion
Ref Expression
yonval (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Proof of Theorem yonval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 yonval.y . 2 𝑌 = (Yon‘𝐶)
2 df-yon 17759 . . 3 Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3 simpr 488 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6721 . . . . . 6 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
5 yonval.o . . . . . 6 𝑂 = (oppCat‘𝐶)
64, 5eqtr4di 2796 . . . . 5 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
73, 6opeq12d 4792 . . . 4 ((𝜑𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
86fveq2d 6721 . . . . 5 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF𝑂))
9 yonval.m . . . . 5 𝑀 = (HomF𝑂)
108, 9eqtr4di 2796 . . . 4 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
117, 10oveq12d 7231 . . 3 ((𝜑𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12 yonval.c . . 3 (𝜑𝐶 ∈ Cat)
13 ovexd 7248 . . 3 (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
142, 11, 12, 13fvmptd2 6826 . 2 (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
151, 14syl5eq 2790 1 (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  cop 4547  cfv 6380  (class class class)co 7213  Catccat 17167  oppCatcoppc 17214   curryF ccurf 17718  HomFchof 17756  Yoncyon 17757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-yon 17759
This theorem is referenced by:  yoncl  17770  yon11  17772  yon12  17773  yon2  17774  yonpropd  17776  oppcyon  17777
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