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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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