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Theorem evlf1 16056
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 2429 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2429 . . . 4  |-  (comp `  D )  =  (comp `  D )
7 eqid 2429 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
81, 2, 3, 4, 5, 6, 7evlfval 16053 . . 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 6333 . . . . 5  |-  ( C 
Func  D )  e.  _V
10 fvex 5891 . . . . . 6  |-  ( Base `  C )  e.  _V
114, 10eqeltri 2513 . . . . 5  |-  B  e. 
_V
129, 11mpt2ex 6884 . . . 4  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
139, 11xpex 6609 . . . . 5  |-  ( ( C  Func  D )  X.  B )  e.  _V
1413, 13mpt2ex 6884 . . . 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 6817 . . 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 17 . 2  |-  ( ph  ->  ( 1st `  E
)  =  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) ) )
17 simprl 762 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
f  =  F )
1817fveq2d 5885 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
19 simprr 764 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
2018, 19fveq12d 5887 . 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 5891 . . 3  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2423a1i 11 . 2  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  _V )
2516, 20, 21, 22, 24ovmpt2d 6438 1  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   [_csb 3401   <.cop 4008    X. cxp 4852   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   Basecbs 15084   Hom chom 15163  compcco 15164   Catccat 15521    Func cfunc 15710   Nat cnat 15797   evalF cevlf 16045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-evlf 16049
This theorem is referenced by:  evlfcllem  16057  evlfcl  16058  uncf1  16072  yonedalem3a  16110  yonedalem3b  16115  yonedainv  16117  yonffthlem  16118
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