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Theorem yonval 16948
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 16938 . . . 4 Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
32a1i 11 . . 3 (𝜑 → Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐)))))
4 simpr 476 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
54fveq2d 6233 . . . . . 6 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
6 yonval.o . . . . . 6 𝑂 = (oppCat‘𝐶)
75, 6syl6eqr 2703 . . . . 5 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
84, 7opeq12d 4441 . . . 4 ((𝜑𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
97fveq2d 6233 . . . . 5 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF𝑂))
10 yonval.m . . . . 5 𝑀 = (HomF𝑂)
119, 10syl6eqr 2703 . . . 4 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
128, 11oveq12d 6708 . . 3 ((𝜑𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
13 yonval.c . . 3 (𝜑𝐶 ∈ Cat)
14 ovexd 6720 . . 3 (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
153, 12, 13, 14fvmptd 6327 . 2 (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
161, 15syl5eq 2697 1 (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216  cmpt 4762  cfv 5926  (class class class)co 6690  Catccat 16372  oppCatcoppc 16418   curryF ccurf 16897  HomFchof 16935  Yoncyon 16936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-yon 16938
This theorem is referenced by:  yoncl  16949  yon11  16951  yon12  16952  yon2  16953  yonpropd  16955  oppcyon  16956
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