MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonval Structured version   Visualization version   GIF version

Theorem yonval 18155
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 18145 . . 3 Yon = (𝑐 ∈ Cat ↦ (βŸ¨π‘, (oppCatβ€˜π‘)⟩ curryF (HomFβ€˜(oppCatβ€˜π‘))))
3 simpr 486 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ 𝑐 = 𝐢)
43fveq2d 6847 . . . . . 6 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (oppCatβ€˜π‘) = (oppCatβ€˜πΆ))
5 yonval.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
64, 5eqtr4di 2791 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (oppCatβ€˜π‘) = 𝑂)
73, 6opeq12d 4839 . . . 4 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ βŸ¨π‘, (oppCatβ€˜π‘)⟩ = ⟨𝐢, π‘‚βŸ©)
86fveq2d 6847 . . . . 5 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (HomFβ€˜(oppCatβ€˜π‘)) = (HomFβ€˜π‘‚))
9 yonval.m . . . . 5 𝑀 = (HomFβ€˜π‘‚)
108, 9eqtr4di 2791 . . . 4 ((πœ‘ ∧ 𝑐 = 𝐢) β†’ (HomFβ€˜(oppCatβ€˜π‘)) = 𝑀)
117, 10oveq12d 7376 . . 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 7393 . . 3 (πœ‘ β†’ (⟨𝐢, π‘‚βŸ© curryF 𝑀) ∈ V)
142, 11, 12, 13fvmptd2 6957 . 2 (πœ‘ β†’ (Yonβ€˜πΆ) = (⟨𝐢, π‘‚βŸ© curryF 𝑀))
151, 14eqtrid 2785 1 (πœ‘ β†’ π‘Œ = (⟨𝐢, π‘‚βŸ© curryF 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444  βŸ¨cop 4593  β€˜cfv 6497  (class class class)co 7358  Catccat 17549  oppCatcoppc 17596   curryF ccurf 18104  HomFchof 18142  Yoncyon 18143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-yon 18145
This theorem is referenced by:  yoncl  18156  yon11  18158  yon12  18159  yon2  18160  yonpropd  18162  oppcyon  18163
  Copyright terms: Public domain W3C validator