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Theorem evlf2 14307
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 14306 . . . 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 6098 . . . . . 6  |-  ( C 
Func  D )  e.  _V
11 fvex 5734 . . . . . . 7  |-  ( Base `  C )  e.  _V
125, 11eqeltri 2505 . . . . . 6  |-  B  e. 
_V
1310, 12mpt2ex 6417 . . . . 5  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
1410, 12xpex 4982 . . . . . 6  |-  ( ( C  Func  D )  X.  B )  e.  _V
1514, 14mpt2ex 6417 . . . . 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 6350 . . . 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 5734 . . . . 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 5724 . . . . 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 6351 . . . . . . 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 2467 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  =  F )
28 fvex 5734 . . . . . 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 5724 . . . . . 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 6351 . . . . . . . 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 2467 . . . . 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 6091 . . . . . 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 5724 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  ( 2nd `  <. F ,  X >. )
)
43 op2ndg 6352 . . . . . . . . . 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 2467 . . . . . . 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 5724 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  ( 2nd `  <. G ,  Y >. )
)
49 op2ndg 6352 . . . . . . . . . 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 2467 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  Y )
5346, 52oveq12d 6091 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) H ( 2nd `  y ) )  =  ( X H Y ) )
5438fveq2d 5724 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  m )  =  ( 1st `  F
) )
5554, 46fveq12d 5726 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  x ) )  =  ( ( 1st `  F
) `  X )
)
5654, 52fveq12d 5726 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  y ) )  =  ( ( 1st `  F
) `  Y )
)
5755, 56opeq12d 3984 . . . . . . . 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 5724 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  n )  =  ( 1st `  G
) )
5958, 52fveq12d 5726 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  n
) `  ( 2nd `  y ) )  =  ( ( 1st `  G
) `  Y )
)
6057, 59oveq12d 6091 . . . . . . 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 5724 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
a `  ( 2nd `  y ) )  =  ( a `  Y
) )
6238fveq2d 5724 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  m )  =  ( 2nd `  F
) )
6362, 46, 52oveq123d 6094 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) ( 2nd `  m
) ( 2nd `  y
) )  =  ( X ( 2nd `  F
) Y ) )
6463fveq1d 5722 . . . . . . 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 6094 . . . . . 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 6128 . . . . 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 3286 . . . 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 3286 . . 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 4902 . . . 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 4902 . . . 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 6098 . . . . 5  |-  ( F N G )  e. 
_V
74 ovex 6098 . . . . 5  |-  ( X H Y )  e. 
_V
7573, 74mpt2ex 6417 . . . 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 6193 . 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 2479 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 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:  evlf2val  14308  evlfcl  14311
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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