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Theorem evlf2 14008
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 14007 . . . 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 5899 . . . . . 6  |-  ( C 
Func  D )  e.  _V
11 fvex 5555 . . . . . . 7  |-  ( Base `  C )  e.  _V
125, 11eqeltri 2366 . . . . . 6  |-  B  e. 
_V
1310, 12mpt2ex 6214 . . . . 5  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
1410, 12xpex 4817 . . . . . 6  |-  ( ( C  Func  D )  X.  B )  e.  _V
1514, 14mpt2ex 6214 . . . . 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 6147 . . . 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 15 . . 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 5555 . . . . 5  |-  ( 1st `  x )  e.  _V
1918a1i 10 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  e.  _V )
20 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  x  =  <. F ,  X >. )
2120fveq2d 5545 . . . . 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 6148 . . . . . . 7  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
2522, 23, 24syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
2625adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st ` 
<. F ,  X >. )  =  F )
2721, 26eqtrd 2328 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  =  F )
28 fvex 5555 . . . . . 6  |-  ( 1st `  y )  e.  _V
2928a1i 10 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  y )  e. 
_V )
30 simplrr 737 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  y  =  <. G ,  Y >. )
3130fveq2d 5545 . . . . . 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 6148 . . . . . . . 8  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  ( 1st `  <. G ,  Y >. )  =  G )
3532, 33, 34syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. G ,  Y >. )  =  G )
3635ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  <. G ,  Y >. )  =  G )
3731, 36eqtrd 2328 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  ->  ( 1st `  y )  =  G )
38 simplr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  m  =  F )
39 simpr 447 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  n  =  G )
4038, 39oveq12d 5892 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
m N n )  =  ( F N G ) )
4120ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  x  =  <. F ,  X >. )
4241fveq2d 5545 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  ( 2nd `  <. F ,  X >. )
)
43 op2ndg 6149 . . . . . . . . . 10  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
4422, 23, 43syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
4544ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  <. F ,  X >. )  =  X )
4642, 45eqtrd 2328 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  X )
4730adantr 451 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  y  =  <. G ,  Y >. )
4847fveq2d 5545 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  ( 2nd `  <. G ,  Y >. )
)
49 op2ndg 6149 . . . . . . . . . 10  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5032, 33, 49syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5150ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  <. G ,  Y >. )  =  Y )
5248, 51eqtrd 2328 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  Y )
5346, 52oveq12d 5892 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) H ( 2nd `  y ) )  =  ( X H Y ) )
5438fveq2d 5545 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  m )  =  ( 1st `  F
) )
5554, 46fveq12d 5547 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  x ) )  =  ( ( 1st `  F
) `  X )
)
5654, 52fveq12d 5547 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  y ) )  =  ( ( 1st `  F
) `  Y )
)
5755, 56opeq12d 3820 . . . . . . . 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 5545 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  n )  =  ( 1st `  G
) )
5958, 52fveq12d 5547 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  n
) `  ( 2nd `  y ) )  =  ( ( 1st `  G
) `  Y )
)
6057, 59oveq12d 5892 . . . . . . 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 5545 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
a `  ( 2nd `  y ) )  =  ( a `  Y
) )
6238fveq2d 5545 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  m )  =  ( 2nd `  F
) )
6362, 46, 52oveq123d 5895 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) ( 2nd `  m
) ( 2nd `  y
) )  =  ( X ( 2nd `  F
) Y ) )
6463fveq1d 5543 . . . . . . 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 5895 . . . . . 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 5926 . . . . 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 3137 . . . 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 3137 . . 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 4737 . . . 4  |-  ( ( F  e.  ( C 
Func  D )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( C  Func  D )  X.  B ) )
7022, 23, 69syl2anc 642 . . 3  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( C  Func  D )  X.  B ) )
71 opelxpi 4737 . . . 4  |-  ( ( G  e.  ( C 
Func  D )  /\  Y  e.  B )  ->  <. G ,  Y >.  e.  ( ( C  Func  D )  X.  B ) )
7232, 33, 71syl2anc 642 . . 3  |-  ( ph  -> 
<. G ,  Y >.  e.  ( ( C  Func  D )  X.  B ) )
73 ovex 5899 . . . . 5  |-  ( F N G )  e. 
_V
74 ovex 5899 . . . . 5  |-  ( X H Y )  e. 
_V
7573, 74mpt2ex 6214 . . . 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 10 . . 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 5991 . 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 2340 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Func cfunc 13744   Nat cnat 13831   evalF cevlf 13999
This theorem is referenced by:  evlf2val  14009  evlfcl  14012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-evlf 14003
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