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Theorem yonval 17108
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 17098 . . . 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 471 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
54fveq2d 6336 . . . . . 6 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
6 yonval.o . . . . . 6 𝑂 = (oppCat‘𝐶)
75, 6syl6eqr 2822 . . . . 5 ((𝜑𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
84, 7opeq12d 4545 . . . 4 ((𝜑𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
97fveq2d 6336 . . . . 5 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF𝑂))
10 yonval.m . . . . 5 𝑀 = (HomF𝑂)
119, 10syl6eqr 2822 . . . 4 ((𝜑𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
128, 11oveq12d 6810 . . 3 ((𝜑𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
13 yonval.c . . 3 (𝜑𝐶 ∈ Cat)
14 ovexd 6824 . . 3 (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
153, 12, 13, 14fvmptd 6430 . 2 (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
161, 15syl5eq 2816 1 (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cop 4320  cmpt 4861  cfv 6031  (class class class)co 6792  Catccat 16531  oppCatcoppc 16577   curryF ccurf 17057  HomFchof 17095  Yoncyon 17096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-yon 17098
This theorem is referenced by:  yoncl  17109  yon11  17111  yon12  17112  yon2  17113  yonpropd  17115  oppcyon  17116
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