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Theorem curfval 14320
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 14311 . . . 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 5742 . . . . 5  |-  ( 1st `  e )  e.  _V
54a1i 11 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  e. 
_V )
6 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  e  =  <. C ,  D >. )
76fveq2d 5732 . . . . 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 6359 . . . . . . 7  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 1st `  <. C ,  D >. )  =  C )
118, 9, 10syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 1st `  <. C ,  D >. )  =  C )
1211adantr 452 . . . . 5  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  <. C ,  D >. )  =  C )
137, 12eqtrd 2468 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  =  C )
14 fvex 5742 . . . . . 6  |-  ( 2nd `  e )  e.  _V
1514a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  e.  _V )
166adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  e  =  <. C ,  D >. )
1716fveq2d 5732 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  ( 2nd `  <. C ,  D >. ) )
18 op2ndg 6360 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 2nd `  <. C ,  D >. )  =  D )
198, 9, 18syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  D >. )  =  D )
2019ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  <. C ,  D >. )  =  D )
2117, 20eqtrd 2468 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  D )
22 simplr 732 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  c  =  C )
2322fveq2d 5732 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  (
Base `  C )
)
24 curfval.a . . . . . . . 8  |-  A  =  ( Base `  C
)
2523, 24syl6eqr 2486 . . . . . . 7  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  A )
26 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  d  =  D )
2726fveq2d 5732 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  (
Base `  D )
)
28 curfval.b . . . . . . . . . 10  |-  B  =  ( Base `  D
)
2927, 28syl6eqr 2486 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  B )
30 simprr 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  f  =  F )
3130ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  f  =  F )
3231fveq2d 5732 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3332oveqd 6098 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
( 1st `  f
) y )  =  ( x ( 1st `  F ) y ) )
3429, 33mpteq12dv 4287 . . . . . . . 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 5732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  (  Hom  `  D )
)
36 curfval.j . . . . . . . . . . . 12  |-  J  =  (  Hom  `  D
)
3735, 36syl6eqr 2486 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  J )
3837oveqd 6098 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y
(  Hom  `  d ) z )  =  ( y J z ) )
3931fveq2d 5732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 2nd `  f )  =  ( 2nd `  F ) )
4039oveqd 6098 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
)  =  ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) )
4122fveq2d 5732 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  ( Id `  C ) )
42 curfval.1 . . . . . . . . . . . . 13  |-  .1.  =  ( Id `  C )
4341, 42syl6eqr 2486 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  .1.  )
4443fveq1d 5730 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  c ) `  x )  =  (  .1.  `  x )
)
45 eqidd 2437 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  g  =  g )
4640, 44, 45oveq123d 6102 . . . . . . . . . 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 4287 . . . . . . . . 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 6136 . . . . . . . 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 3992 . . . . . . 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 4287 . . . . . 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 5732 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  (  Hom  `  C )
)
52 curfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
5351, 52syl6eqr 2486 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  H )
5453oveqd 6098 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
(  Hom  `  c ) y )  =  ( x H y ) )
5539oveqd 6098 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  z >. ( 2nd `  f ) <.
y ,  z >.
)  =  ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) )
5626fveq2d 5732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  ( Id `  D ) )
57 curfval.i . . . . . . . . . . . 12  |-  I  =  ( Id `  D
)
5856, 57syl6eqr 2486 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  I )
5958fveq1d 5730 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  d ) `  z )  =  ( I `  z ) )
6055, 45, 59oveq123d 6102 . . . . . . . . 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 4287 . . . . . . . 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 4287 . . . . . . 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 6136 . . . . . 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 3992 . . . . 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 3294 . . . 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 3294 . . 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 4427 . . . 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 2964 . . . 4  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  F  e.  _V )
7169, 70syl 16 . . 3  |-  ( ph  ->  F  e.  _V )
72 opex 4427 . . . 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 6201 . 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 2480 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251   <.cop 3817    e. cmpt 4266   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   Basecbs 13469    Hom chom 13540   Catccat 13889   Idccid 13890    Func cfunc 14051    X.c cxpc 14265   curryF ccurf 14307
This theorem is referenced by:  curf1fval  14321  curf2  14326  curfcl  14329  curfpropd  14330  curfuncf  14335
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-curf 14311
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