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Theorem yonval 18253
Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yonval.y π‘Œ = (Yonβ€˜πΆ)
yonval.c (πœ‘ β†’ 𝐢 ∈ Cat)
yonval.o 𝑂 = (oppCatβ€˜πΆ)
yonval.m 𝑀 = (HomFβ€˜π‘‚)
Assertion
Ref Expression
yonval (πœ‘ β†’ π‘Œ = (⟨𝐢, π‘‚βŸ© curryF 𝑀))

Proof of Theorem yonval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 yonval.y . 2 π‘Œ = (Yonβ€˜πΆ)
2 df-yon 18243 . . 3 Yon = (𝑐 ∈ Cat ↦ (βŸ¨π‘, (oppCatβ€˜π‘)⟩ curryF (HomFβ€˜(oppCatβ€˜π‘))))
3 simpr 484 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ 𝑐 = 𝐢)
43fveq2d 6901 . . . . . 6 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (oppCatβ€˜π‘) = (oppCatβ€˜πΆ))
5 yonval.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
64, 5eqtr4di 2786 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (oppCatβ€˜π‘) = 𝑂)
73, 6opeq12d 4882 . . . 4 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ βŸ¨π‘, (oppCatβ€˜π‘)⟩ = ⟨𝐢, π‘‚βŸ©)
86fveq2d 6901 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (HomFβ€˜(oppCatβ€˜π‘)) = (HomFβ€˜π‘‚))
9 yonval.m . . . . 5 𝑀 = (HomFβ€˜π‘‚)
108, 9eqtr4di 2786 . . . 4 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (HomFβ€˜(oppCatβ€˜π‘)) = 𝑀)
117, 10oveq12d 7438 . . 3 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (βŸ¨π‘, (oppCatβ€˜π‘)⟩ curryF (HomFβ€˜(oppCatβ€˜π‘))) = (⟨𝐢, π‘‚βŸ© curryF 𝑀))
12 yonval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
13 ovexd 7455 . . 3 (πœ‘ β†’ (⟨𝐢, π‘‚βŸ© curryF 𝑀) ∈ V)
142, 11, 12, 13fvmptd2 7013 . 2 (πœ‘ β†’ (Yonβ€˜πΆ) = (⟨𝐢, π‘‚βŸ© curryF 𝑀))
151, 14eqtrid 2780 1 (πœ‘ β†’ π‘Œ = (⟨𝐢, π‘‚βŸ© curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471  βŸ¨cop 4635  β€˜cfv 6548  (class class class)co 7420  Catccat 17644  oppCatcoppc 17691   curryF ccurf 18202  HomFchof 18240  Yoncyon 18241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-yon 18243
This theorem is referenced by:  yoncl  18254  yon11  18256  yon12  18257  yon2  18258  yonpropd  18260  oppcyon  18261
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