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Theorem evlf1 15347
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 2467 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2467 . . . 4  |-  (comp `  D )  =  (comp `  D )
7 eqid 2467 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
81, 2, 3, 4, 5, 6, 7evlfval 15344 . . 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 6309 . . . . 5  |-  ( C 
Func  D )  e.  _V
10 fvex 5876 . . . . . 6  |-  ( Base `  C )  e.  _V
114, 10eqeltri 2551 . . . . 5  |-  B  e. 
_V
129, 11mpt2ex 6860 . . . 4  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
139, 11xpex 6588 . . . . 5  |-  ( ( C  Func  D )  X.  B )  e.  _V
1413, 13mpt2ex 6860 . . . 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 6794 . . 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 755 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
f  =  F )
1817fveq2d 5870 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
19 simprr 756 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
2018, 19fveq12d 5872 . 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 5876 . . 3  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2423a1i 11 . 2  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  _V )
2516, 20, 21, 22, 24ovmpt2d 6414 1  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435   <.cop 4033    X. cxp 4997   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919    Func cfunc 15081   Nat cnat 15168   evalF cevlf 15336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-evlf 15340
This theorem is referenced by:  evlfcllem  15348  evlfcl  15349  uncf1  15363  yonedalem3a  15401  yonedalem3b  15406  yonedainv  15408  yonffthlem  15409
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