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Theorem evlf2 14242
Description: Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e  |-  E  =  ( C evalF  D )
evlfval.c  |-  ( ph  ->  C  e.  Cat )
evlfval.d  |-  ( ph  ->  D  e.  Cat )
evlfval.b  |-  B  =  ( Base `  C
)
evlfval.h  |-  H  =  (  Hom  `  C
)
evlfval.o  |-  .x.  =  (comp `  D )
evlfval.n  |-  N  =  ( C Nat  D )
evlf2.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
evlf2.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
evlf2.x  |-  ( ph  ->  X  e.  B )
evlf2.y  |-  ( ph  ->  Y  e.  B )
evlf2.l  |-  L  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
Assertion
Ref Expression
evlf2  |-  ( ph  ->  L  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  g ) ) ) )
Distinct variable groups:    g, a, C    D, a, g    g, H    F, a, g    N, a, g    G, a, g    ph, a, g    .x. , a,
g    X, a, g    Y, a, g
Allowed substitution hints:    B( g, a)    E( g, a)    H( a)    L( g, a)

Proof of Theorem evlf2
Dummy variables  f  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlf2.l . 2  |-  L  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
2 evlfval.e . . . . 5  |-  E  =  ( C evalF  D )
3 evlfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
4 evlfval.d . . . . 5  |-  ( ph  ->  D  e.  Cat )
5 evlfval.b . . . . 5  |-  B  =  ( Base `  C
)
6 evlfval.h . . . . 5  |-  H  =  (  Hom  `  C
)
7 evlfval.o . . . . 5  |-  .x.  =  (comp `  D )
8 evlfval.n . . . . 5  |-  N  =  ( C Nat  D )
92, 3, 4, 5, 6, 7, 8evlfval 14241 . . . 4  |-  ( 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 N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>. )
10 ovex 6045 . . . . . 6  |-  ( C 
Func  D )  e.  _V
11 fvex 5682 . . . . . . 7  |-  ( Base `  C )  e.  _V
125, 11eqeltri 2457 . . . . . 6  |-  B  e. 
_V
1310, 12mpt2ex 6364 . . . . 5  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
1410, 12xpex 4930 . . . . . 6  |-  ( ( C  Func  D )  X.  B )  e.  _V
1514, 14mpt2ex 6364 . . . . 5  |-  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )  e.  _V
1613, 15op2ndd 6297 . . . 4  |-  ( 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 N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>.  ->  ( 2nd `  E
)  =  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) ) )
179, 16syl 16 . . 3  |-  ( ph  ->  ( 2nd `  E
)  =  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) ) )
18 fvex 5682 . . . . 5  |-  ( 1st `  x )  e.  _V
1918a1i 11 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  e.  _V )
20 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  x  =  <. F ,  X >. )
2120fveq2d 5672 . . . . 5  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. F ,  X >. ) )
22 evlf2.f . . . . . . 7  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
23 evlf2.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
24 op1stg 6298 . . . . . . 7  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
2522, 23, 24syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
2625adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st ` 
<. F ,  X >. )  =  F )
2721, 26eqtrd 2419 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  =  F )
28 fvex 5682 . . . . . 6  |-  ( 1st `  y )  e.  _V
2928a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  y )  e. 
_V )
30 simplrr 738 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  y  =  <. G ,  Y >. )
3130fveq2d 5672 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  y )  =  ( 1st `  <. G ,  Y >. )
)
32 evlf2.g . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
33 evlf2.y . . . . . . . 8  |-  ( ph  ->  Y  e.  B )
34 op1stg 6298 . . . . . . . 8  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  ( 1st `  <. G ,  Y >. )  =  G )
3532, 33, 34syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. G ,  Y >. )  =  G )
3635ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  <. G ,  Y >. )  =  G )
3731, 36eqtrd 2419 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  y )  =  G )
38 simplr 732 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  m  =  F )
39 simpr 448 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  n  =  G )
4038, 39oveq12d 6038 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
m N n )  =  ( F N G ) )
4120ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  x  =  <. F ,  X >. )
4241fveq2d 5672 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  ( 2nd `  <. F ,  X >. )
)
43 op2ndg 6299 . . . . . . . . . 10  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
4422, 23, 43syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
4544ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  <. F ,  X >. )  =  X )
4642, 45eqtrd 2419 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  X )
4730adantr 452 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  y  =  <. G ,  Y >. )
4847fveq2d 5672 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  ( 2nd `  <. G ,  Y >. )
)
49 op2ndg 6299 . . . . . . . . . 10  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5032, 33, 49syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5150ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5248, 51eqtrd 2419 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  Y )
5346, 52oveq12d 6038 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) H ( 2nd `  y ) )  =  ( X H Y ) )
5438fveq2d 5672 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  m )  =  ( 1st `  F
) )
5554, 46fveq12d 5674 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  x ) )  =  ( ( 1st `  F
) `  X )
)
5654, 52fveq12d 5674 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  y ) )  =  ( ( 1st `  F
) `  Y )
)
5755, 56opeq12d 3934 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  <. (
( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
)
5839fveq2d 5672 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  n )  =  ( 1st `  G
) )
5958, 52fveq12d 5674 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  n
) `  ( 2nd `  y ) )  =  ( ( 1st `  G
) `  Y )
)
6057, 59oveq12d 6038 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( <. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.  .x.  ( ( 1st `  n
) `  ( 2nd `  y ) ) )  =  ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) )
6152fveq2d 5672 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
a `  ( 2nd `  y ) )  =  ( a `  Y
) )
6238fveq2d 5672 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  m )  =  ( 2nd `  F
) )
6362, 46, 52oveq123d 6041 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) ( 2nd `  m
) ( 2nd `  y
) )  =  ( X ( 2nd `  F
) Y ) )
6463fveq1d 5670 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( ( 2nd `  x
) ( 2nd `  m
) ( 2nd `  y
) ) `  g
)  =  ( ( X ( 2nd `  F
) Y ) `  g ) )
6560, 61, 64oveq123d 6041 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) )  =  ( ( a `  Y
) ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) ) )
6640, 53, 65mpt2eq123dv 6075 . . . . 5  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.  .x.  ( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) )  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `
 Y ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) ) ) )
6729, 37, 66csbied2 3237 . . . 4  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) )  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y ) 
|->  ( ( a `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) ) ) )
6819, 27, 67csbied2 3237 . . 3  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) )  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y ) 
|->  ( ( a `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) ) ) )
69 opelxpi 4850 . . . 4  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( C  Func  D )  X.  B ) )
7022, 23, 69syl2anc 643 . . 3  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( C  Func  D )  X.  B ) )
71 opelxpi 4850 . . . 4  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  <. G ,  Y >.  e.  ( ( C  Func  D )  X.  B ) )
7232, 33, 71syl2anc 643 . . 3  |-  ( ph  -> 
<. G ,  Y >.  e.  ( ( C  Func  D )  X.  B ) )
73 ovex 6045 . . . . 5  |-  ( F N G )  e. 
_V
74 ovex 6045 . . . . 5  |-  ( X H Y )  e. 
_V
7573, 74mpt2ex 6364 . . . 4  |-  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  g ) ) )  e.  _V
7675a1i 11 . . 3  |-  ( ph  ->  ( a  e.  ( F N G ) ,  g  e.  ( X H Y ) 
|->  ( ( a `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) ) )  e.  _V )
7717, 68, 70, 72, 76ovmpt2d 6140 . 2  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  g ) ) ) )
781, 77syl5eq 2431 1  |-  ( ph  ->  L  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  g ) ) ) )
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:  evlf2val  14243  evlfcl  14246
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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