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Theorem curfval 15618
Description: Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
curfval.g  |-  G  =  ( <. C ,  D >. curryF  F
)
curfval.a  |-  A  =  ( Base `  C
)
curfval.c  |-  ( ph  ->  C  e.  Cat )
curfval.d  |-  ( ph  ->  D  e.  Cat )
curfval.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
curfval.b  |-  B  =  ( Base `  D
)
curfval.j  |-  J  =  ( Hom  `  D
)
curfval.1  |-  .1.  =  ( Id `  C )
curfval.h  |-  H  =  ( Hom  `  C
)
curfval.i  |-  I  =  ( Id `  D
)
Assertion
Ref Expression
curfval  |-  ( ph  ->  G  =  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
Distinct variable groups:    x, g,
y, z,  .1.    x, A, y    B, g, x, y, z    C, g, x, y, z    D, g, x, y, z    g, H, y, z    ph, g, x, y, z    g, E, y, z    g, J, x   
g, F, x, y, z
Allowed substitution hints:    A( z, g)    E( x)    G( x, y, z, g)    H( x)    I( x, y, z, g)    J( y, z)

Proof of Theorem curfval
Dummy variables  c 
d  e  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . 2  |-  G  =  ( <. C ,  D >. curryF  F
)
2 df-curf 15609 . . . 4  |- curryF  =  ( e  e. 
_V ,  f  e. 
_V  |->  [_ ( 1st `  e
)  /  c ]_ [_ ( 2nd `  e
)  /  d ]_ <. ( x  e.  (
Base `  c )  |-> 
<. ( y  e.  (
Base `  d )  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  ( Base `  d
) ,  z  e.  ( Base `  d
)  |->  ( g  e.  ( y ( Hom  `  d ) z ) 
|->  ( ( ( Id
`  c ) `  x ) ( <.
x ,  y >.
( 2nd `  f
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) ) >. )
32a1i 11 . . 3  |-  ( ph  -> curryF  =  ( e  e.  _V ,  f  e.  _V  |->  [_ ( 1st `  e
)  /  c ]_ [_ ( 2nd `  e
)  /  d ]_ <. ( x  e.  (
Base `  c )  |-> 
<. ( y  e.  (
Base `  d )  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  ( Base `  d
) ,  z  e.  ( Base `  d
)  |->  ( g  e.  ( y ( Hom  `  d ) z ) 
|->  ( ( ( Id
`  c ) `  x ) ( <.
x ,  y >.
( 2nd `  f
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) ) >. )
)
4 fvex 5882 . . . . 5  |-  ( 1st `  e )  e.  _V
54a1i 11 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  e. 
_V )
6 simprl 756 . . . . . 6  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  e  =  <. C ,  D >. )
76fveq2d 5876 . . . . 5  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  =  ( 1st `  <. C ,  D >. )
)
8 curfval.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
9 curfval.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
10 op1stg 6811 . . . . . . 7  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 1st `  <. C ,  D >. )  =  C )
118, 9, 10syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st `  <. C ,  D >. )  =  C )
1211adantr 465 . . . . 5  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  <. C ,  D >. )  =  C )
137, 12eqtrd 2498 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  =  C )
14 fvex 5882 . . . . . 6  |-  ( 2nd `  e )  e.  _V
1514a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  e.  _V )
166adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  e  =  <. C ,  D >. )
1716fveq2d 5876 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  ( 2nd `  <. C ,  D >. ) )
18 op2ndg 6812 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 2nd `  <. C ,  D >. )  =  D )
198, 9, 18syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  D >. )  =  D )
2019ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  <. C ,  D >. )  =  D )
2117, 20eqtrd 2498 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  D )
22 simplr 755 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  c  =  C )
2322fveq2d 5876 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  (
Base `  C )
)
24 curfval.a . . . . . . . 8  |-  A  =  ( Base `  C
)
2523, 24syl6eqr 2516 . . . . . . 7  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  A )
26 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  d  =  D )
2726fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  (
Base `  D )
)
28 curfval.b . . . . . . . . . 10  |-  B  =  ( Base `  D
)
2927, 28syl6eqr 2516 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  B )
30 simprr 757 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  f  =  F )
3130ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  f  =  F )
3231fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3332oveqd 6313 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
( 1st `  f
) y )  =  ( x ( 1st `  F ) y ) )
3429, 33mpteq12dv 4535 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y  e.  ( Base `  d
)  |->  ( x ( 1st `  f ) y ) )  =  ( y  e.  B  |->  ( x ( 1st `  F ) y ) ) )
3526fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Hom  `  d )  =  ( Hom  `  D )
)
36 curfval.j . . . . . . . . . . . 12  |-  J  =  ( Hom  `  D
)
3735, 36syl6eqr 2516 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Hom  `  d )  =  J )
3837oveqd 6313 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y
( Hom  `  d ) z )  =  ( y J z ) )
3931fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 2nd `  f )  =  ( 2nd `  F ) )
4039oveqd 6313 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
)  =  ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) )
4122fveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  ( Id `  C ) )
42 curfval.1 . . . . . . . . . . . . 13  |-  .1.  =  ( Id `  C )
4341, 42syl6eqr 2516 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  .1.  )
4443fveq1d 5874 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  c ) `  x )  =  (  .1.  `  x )
)
45 eqidd 2458 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  g  =  g )
4640, 44, 45oveq123d 6317 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( (
( Id `  c
) `  x )
( <. x ,  y
>. ( 2nd `  f
) <. x ,  z
>. ) g )  =  ( (  .1.  `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) )
4738, 46mpteq12dv 4535 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( g  e.  ( y ( Hom  `  d ) z ) 
|->  ( ( ( Id
`  c ) `  x ) ( <.
x ,  y >.
( 2nd `  f
) <. x ,  z
>. ) g ) )  =  ( g  e.  ( y J z )  |->  ( (  .1.  `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) )
4829, 29, 47mpt2eq123dv 6358 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y  e.  ( Base `  d
) ,  z  e.  ( Base `  d
)  |->  ( g  e.  ( y ( Hom  `  d ) z ) 
|->  ( ( ( Id
`  c ) `  x ) ( <.
x ,  y >.
( 2nd `  f
) <. x ,  z
>. ) g ) ) )  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) )
4934, 48opeq12d 4227 . . . . . . 7  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  <. ( y  e.  ( Base `  d
)  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  (
Base `  d ) ,  z  e.  ( Base `  d )  |->  ( g  e.  ( y ( Hom  `  d
) z )  |->  ( ( ( Id `  c ) `  x
) ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
) g ) ) ) >.  =  <. ( y  e.  B  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. )
5025, 49mpteq12dv 4535 . . . . . 6  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x  e.  ( Base `  c
)  |->  <. ( y  e.  ( Base `  d
)  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  (
Base `  d ) ,  z  e.  ( Base `  d )  |->  ( g  e.  ( y ( Hom  `  d
) z )  |->  ( ( ( Id `  c ) `  x
) ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
) g ) ) ) >. )  =  ( x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) )
5122fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Hom  `  c )  =  ( Hom  `  C )
)
52 curfval.h . . . . . . . . . 10  |-  H  =  ( Hom  `  C
)
5351, 52syl6eqr 2516 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Hom  `  c )  =  H )
5453oveqd 6313 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
( Hom  `  c ) y )  =  ( x H y ) )
5539oveqd 6313 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  z >. ( 2nd `  f ) <.
y ,  z >.
)  =  ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) )
5626fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  ( Id `  D ) )
57 curfval.i . . . . . . . . . . . 12  |-  I  =  ( Id `  D
)
5856, 57syl6eqr 2516 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  I )
5958fveq1d 5874 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  d ) `  z )  =  ( I `  z ) )
6055, 45, 59oveq123d 6317 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( g
( <. x ,  z
>. ( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) )  =  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) )
6129, 60mpteq12dv 4535 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( z  e.  ( Base `  d
)  |->  ( g (
<. x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) )  =  ( z  e.  B  |->  ( g (
<. x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )
6254, 61mpteq12dv 4535 . . . . . . 7  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) )  =  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) )
6325, 25, 62mpt2eq123dv 6358 . . . . . 6  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x  e.  ( Base `  c
) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) )  =  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) )
6450, 63opeq12d 4227 . . . . 5  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  <. ( x  e.  ( Base `  c
)  |->  <. ( y  e.  ( Base `  d
)  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  (
Base `  d ) ,  z  e.  ( Base `  d )  |->  ( g  e.  ( y ( Hom  `  d
) z )  |->  ( ( ( Id `  c ) `  x
) ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) ) >.  =  <. ( x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
6515, 21, 64csbied2 3458 . . . 4  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  [_ ( 2nd `  e
)  /  d ]_ <. ( x  e.  (
Base `  c )  |-> 
<. ( y  e.  (
Base `  d )  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  ( Base `  d
) ,  z  e.  ( Base `  d
)  |->  ( g  e.  ( y ( Hom  `  d ) z ) 
|->  ( ( ( Id
`  c ) `  x ) ( <.
x ,  y >.
( 2nd `  f
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) ) >.  =  <. ( x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
665, 13, 65csbied2 3458 . . 3  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  [_ ( 1st `  e )  / 
c ]_ [_ ( 2nd `  e )  /  d ]_ <. ( x  e.  ( Base `  c
)  |->  <. ( y  e.  ( Base `  d
)  |->  ( x ( 1st `  f ) y ) ) ,  ( y  e.  (
Base `  d ) ,  z  e.  ( Base `  d )  |->  ( g  e.  ( y ( Hom  `  d
) z )  |->  ( ( ( Id `  c ) `  x
) ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( x ( Hom  `  c ) y ) 
|->  ( z  e.  (
Base `  d )  |->  ( g ( <.
x ,  z >.
( 2nd `  f
) <. y ,  z
>. ) ( ( Id
`  d ) `  z ) ) ) ) ) >.  =  <. ( x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
67 opex 4720 . . . 4  |-  <. C ,  D >.  e.  _V
6867a1i 11 . . 3  |-  ( ph  -> 
<. C ,  D >.  e. 
_V )
69 curfval.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
70 elex 3118 . . . 4  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  F  e.  _V )
7169, 70syl 16 . . 3  |-  ( ph  ->  F  e.  _V )
72 opex 4720 . . . 4  |-  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >.  e.  _V
7372a1i 11 . . 3  |-  ( ph  -> 
<. ( x  e.  A  |-> 
<. ( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >.  e.  _V )
743, 66, 68, 71, 73ovmpt2d 6429 . 2  |-  ( ph  ->  ( <. C ,  D >. curryF  F
)  =  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
751, 74syl5eq 2510 1  |-  ( ph  ->  G  =  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   [_csb 3430   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   Basecbs 14643   Hom chom 14722   Catccat 15080   Idccid 15081    Func cfunc 15269    X.c cxpc 15563   curryF ccurf 15605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-curf 15609
This theorem is referenced by:  curf1fval  15619  curf2  15624  curfcl  15627  curfpropd  15628  curfuncf  15633
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