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Theorem evlf1 14244
Description: Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlf1.e  |-  E  =  ( C evalF  D )
evlf1.c  |-  ( ph  ->  C  e.  Cat )
evlf1.d  |-  ( ph  ->  D  e.  Cat )
evlf1.b  |-  B  =  ( Base `  C
)
evlf1.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
evlf1.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
evlf1  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )

Proof of Theorem evlf1
Dummy variables  x  y  f  a  g  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlf1.e . . . 4  |-  E  =  ( C evalF  D )
2 evlf1.c . . . 4  |-  ( ph  ->  C  e.  Cat )
3 evlf1.d . . . 4  |-  ( ph  ->  D  e.  Cat )
4 evlf1.b . . . 4  |-  B  =  ( Base `  C
)
5 eqid 2387 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2387 . . . 4  |-  (comp `  D )  =  (comp `  D )
7 eqid 2387 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
81, 2, 3, 4, 5, 6, 7evlfval 14241 . . 3  |-  ( ph  ->  E  =  <. (
f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m ( C Nat  D
) n ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  D )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) ) >. )
9 ovex 6045 . . . . 5  |-  ( C 
Func  D )  e.  _V
10 fvex 5682 . . . . . 6  |-  ( Base `  C )  e.  _V
114, 10eqeltri 2457 . . . . 5  |-  B  e. 
_V
129, 11mpt2ex 6364 . . . 4  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
139, 11xpex 4930 . . . . 5  |-  ( ( C  Func  D )  X.  B )  e.  _V
1413, 13mpt2ex 6364 . . . 4  |-  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m ( C Nat  D
) n ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  D )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) )  e.  _V
1512, 14op1std 6296 . . 3  |-  ( E  =  <. ( f  e.  ( C  Func  D
) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C 
Func  D )  X.  B
)  |->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m ( C Nat  D
) n ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  D )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) ) >.  ->  ( 1st `  E )  =  ( f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) )
168, 15syl 16 . 2  |-  ( ph  ->  ( 1st `  E
)  =  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) ) )
17 simprl 733 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
f  =  F )
1817fveq2d 5672 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
19 simprr 734 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
2018, 19fveq12d 5674 . 2  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  X
) )
21 evlf1.f . 2  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
22 evlf1.x . 2  |-  ( ph  ->  X  e.  B )
23 fvex 5682 . . 3  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2423a1i 11 . 2  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  _V )
2516, 20, 21, 22, 24ovmpt2d 6140 1  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   [_csb 3194   <.cop 3760    X. cxp 4816   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287   Basecbs 13396    Hom chom 13467  compcco 13468   Catccat 13816    Func cfunc 13978   Nat cnat 14065   evalF cevlf 14233
This theorem is referenced by:  evlfcllem  14245  evlfcl  14246  uncf1  14260  yonedalem3a  14298  yonedalem3b  14303  yonedainv  14305  yonffthlem  14306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-evlf 14237
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