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Theorem evlf2 14270
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 14269 . . . 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 6065 . . . . . 6  |-  ( C 
Func  D )  e.  _V
11 fvex 5701 . . . . . . 7  |-  ( Base `  C )  e.  _V
125, 11eqeltri 2474 . . . . . 6  |-  B  e. 
_V
1310, 12mpt2ex 6384 . . . . 5  |-  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) )  e.  _V
1410, 12xpex 4949 . . . . . 6  |-  ( ( C  Func  D )  X.  B )  e.  _V
1514, 14mpt2ex 6384 . . . . 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 6317 . . . 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 5701 . . . . 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 5691 . . . . 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 6318 . . . . . . 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 2436 . . . 4  |-  ( (
ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. )
)  ->  ( 1st `  x )  =  F )
28 fvex 5701 . . . . . 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 5691 . . . . . 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 6318 . . . . . . . 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 2436 . . . . 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 6058 . . . . . 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 5691 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  x )  =  ( 2nd `  <. F ,  X >. )
)
43 op2ndg 6319 . . . . . . . . . 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 2436 . . . . . . 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 5691 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  ( 2nd `  <. G ,  Y >. )
)
49 op2ndg 6319 . . . . . . . . . 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 2436 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  y )  =  Y )
5346, 52oveq12d 6058 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) H ( 2nd `  y ) )  =  ( X H Y ) )
5438fveq2d 5691 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  m )  =  ( 1st `  F
) )
5554, 46fveq12d 5693 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  x ) )  =  ( ( 1st `  F
) `  X )
)
5654, 52fveq12d 5693 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  m
) `  ( 2nd `  y ) )  =  ( ( 1st `  F
) `  Y )
)
5755, 56opeq12d 3952 . . . . . . . 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 5691 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 1st `  n )  =  ( 1st `  G
) )
5958, 52fveq12d 5693 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 1st `  n
) `  ( 2nd `  y ) )  =  ( ( 1st `  G
) `  Y )
)
6057, 59oveq12d 6058 . . . . . . 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 5691 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
a `  ( 2nd `  y ) )  =  ( a `  Y
) )
6238fveq2d 5691 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  ( 2nd `  m )  =  ( 2nd `  F
) )
6362, 46, 52oveq123d 6061 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  =  <. F ,  X >.  /\  y  =  <. G ,  Y >. ) )  /\  m  =  F )  /\  n  =  G )  ->  (
( 2nd `  x
) ( 2nd `  m
) ( 2nd `  y
) )  =  ( X ( 2nd `  F
) Y ) )
6463fveq1d 5689 . . . . . . 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 6061 . . . . . 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 6095 . . . . 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 3254 . . . 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 3254 . . 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 4869 . . . 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 4869 . . . 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 6065 . . . . 5  |-  ( F N G )  e. 
_V
74 ovex 6065 . . . . 5  |-  ( X H Y )  e. 
_V
7573, 74mpt2ex 6384 . . . 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 6160 . 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 2448 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 1721   _Vcvv 2916   [_csb 3211   <.cop 3777    X. cxp 4835   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844    Func cfunc 14006   Nat cnat 14093   evalF cevlf 14261
This theorem is referenced by:  evlf2val  14271  evlfcl  14274
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-evlf 14265
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