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Theorem yonval 15092
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 15082 . . . 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 5716 . . . . . 6  |-  ( (
ph  /\  c  =  C )  ->  (oppCat `  c )  =  (oppCat `  C ) )
6 yonval.o . . . . . 6  |-  O  =  (oppCat `  C )
75, 6syl6eqr 2493 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  (oppCat `  c )  =  O )
84, 7opeq12d 4088 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  <. c ,  (oppCat `  c ) >.  =  <. C ,  O >. )
97fveq2d 5716 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  (HomF `  (oppCat `  c ) )  =  (HomF
`  O ) )
10 yonval.m . . . . 5  |-  M  =  (HomF
`  O )
119, 10syl6eqr 2493 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  (HomF `  (oppCat `  c ) )  =  M )
128, 11oveq12d 6130 . . 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 6137 . . . 4  |-  ( <. C ,  O >. curryF  M )  e.  _V
1514a1i 11 . . 3  |-  ( ph  ->  ( <. C ,  O >. curryF  M
)  e.  _V )
163, 12, 13, 15fvmptd 5800 . 2  |-  ( ph  ->  (Yon `  C )  =  ( <. C ,  O >. curryF  M ) )
171, 16syl5eq 2487 1  |-  ( ph  ->  Y  =  ( <. C ,  O >. curryF  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993   <.cop 3904    e. cmpt 4371   ` cfv 5439  (class class class)co 6112   Catccat 14623  oppCatcoppc 14671   curryF ccurf 15041  HomFchof 15079  Yoncyon 15080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-yon 15082
This theorem is referenced by:  yoncl  15093  yon11  15095  yon12  15096  yon2  15097  yonpropd  15099  oppcyon  15100
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