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Theorem yonval 15379
Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yonval.y  |-  Y  =  (Yon `  C )
yonval.c  |-  ( ph  ->  C  e.  Cat )
yonval.o  |-  O  =  (oppCat `  C )
yonval.m  |-  M  =  (HomF
`  O )
Assertion
Ref Expression
yonval  |-  ( ph  ->  Y  =  ( <. 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 15369 . . . 4  |- Yon  =  ( c  e.  Cat  |->  (
<. c ,  (oppCat `  c ) >. curryF  (HomF
`  (oppCat `  c )
) ) )
32a1i 11 . . 3  |-  ( ph  -> Yon  =  ( c  e. 
Cat  |->  ( <. c ,  (oppCat `  c ) >. curryF  (HomF `  (oppCat `  c ) ) ) ) )
4 simpr 461 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  c  =  C )
54fveq2d 5863 . . . . . 6  |-  ( (
ph  /\  c  =  C )  ->  (oppCat `  c )  =  (oppCat `  C ) )
6 yonval.o . . . . . 6  |-  O  =  (oppCat `  C )
75, 6syl6eqr 2521 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  (oppCat `  c )  =  O )
84, 7opeq12d 4216 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  <. c ,  (oppCat `  c ) >.  =  <. C ,  O >. )
97fveq2d 5863 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  (HomF `  (oppCat `  c ) )  =  (HomF
`  O ) )
10 yonval.m . . . . 5  |-  M  =  (HomF
`  O )
119, 10syl6eqr 2521 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  (HomF `  (oppCat `  c ) )  =  M )
128, 11oveq12d 6295 . . 3  |-  ( (
ph  /\  c  =  C )  ->  ( <. c ,  (oppCat `  c ) >. curryF  (HomF
`  (oppCat `  c )
) )  =  (
<. C ,  O >. curryF  M ) )
13 yonval.c . . 3  |-  ( ph  ->  C  e.  Cat )
14 ovex 6302 . . . 4  |-  ( <. C ,  O >. curryF  M )  e.  _V
1514a1i 11 . . 3  |-  ( ph  ->  ( <. C ,  O >. curryF  M
)  e.  _V )
163, 12, 13, 15fvmptd 5948 . 2  |-  ( ph  ->  (Yon `  C )  =  ( <. C ,  O >. curryF  M ) )
171, 16syl5eq 2515 1  |-  ( ph  ->  Y  =  ( <. C ,  O >. curryF  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   Catccat 14910  oppCatcoppc 14958   curryF ccurf 15328  HomFchof 15366  Yoncyon 15367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-yon 15369
This theorem is referenced by:  yoncl  15380  yon11  15382  yon12  15383  yon2  15384  yonpropd  15386  oppcyon  15387
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