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Theorem yonval 18318
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 18308 . . 3 Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3 simpr 484 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6911 . . . . . 6 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
5 yonval.o . . . . . 6 𝑂 = (oppCat‘𝐶)
64, 5eqtr4di 2793 . . . . 5 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
73, 6opeq12d 4886 . . . 4 ((𝜑𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
86fveq2d 6911 . . . . 5 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF𝑂))
9 yonval.m . . . . 5 𝑀 = (HomF𝑂)
108, 9eqtr4di 2793 . . . 4 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
117, 10oveq12d 7449 . . 3 ((𝜑𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12 yonval.c . . 3 (𝜑𝐶 ∈ Cat)
13 ovexd 7466 . . 3 (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
142, 11, 12, 13fvmptd2 7024 . 2 (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
151, 14eqtrid 2787 1 (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cfv 6563  (class class class)co 7431  Catccat 17709  oppCatcoppc 17756   curryF ccurf 18267  HomFchof 18305  Yoncyon 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-yon 18308
This theorem is referenced by:  yoncl  18319  yon11  18321  yon12  18322  yon2  18323  yonpropd  18325  oppcyon  18326
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