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Theorem evlf1 14309
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 2435 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2435 . . . 4  |-  (comp `  D )  =  (comp `  D )
7 eqid 2435 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
81, 2, 3, 4, 5, 6, 7evlfval 14306 . . 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 6098 . . . . 5  |-  ( C 
Func  D )  e.  _V
10 fvex 5734 . . . . . 6  |-  ( Base `  C )  e.  _V
114, 10eqeltri 2505 . . . . 5  |-  B  e. 
_V
129, 11mpt2ex 6417 . . . 4  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
139, 11xpex 4982 . . . . 5  |-  ( ( C  Func  D )  X.  B )  e.  _V
1413, 13mpt2ex 6417 . . . 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 6349 . . 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 5724 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
19 simprr 734 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
2018, 19fveq12d 5726 . 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 5734 . . 3  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2423a1i 11 . 2  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  _V )
2516, 20, 21, 22, 24ovmpt2d 6193 1  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243   <.cop 3809    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881    Func cfunc 14043   Nat cnat 14130   evalF cevlf 14298
This theorem is referenced by:  evlfcllem  14310  evlfcl  14311  uncf1  14325  yonedalem3a  14363  yonedalem3b  14368  yonedainv  14370  yonffthlem  14371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-evlf 14302
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