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Theorem curfval 13997
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 13988 . . . 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 10 . . 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 5539 . . . . 5  |-  ( 1st `  e )  e.  _V
54a1i 10 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  e. 
_V )
6 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  e  =  <. C ,  D >. )
76fveq2d 5529 . . . . 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 6132 . . . . . . 7  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 1st `  <. C ,  D >. )  =  C )
118, 9, 10syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 1st `  <. C ,  D >. )  =  C )
1211adantr 451 . . . . 5  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  <. C ,  D >. )  =  C )
137, 12eqtrd 2315 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  =  C )
14 fvex 5539 . . . . . 6  |-  ( 2nd `  e )  e.  _V
1514a1i 10 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  e.  _V )
166adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  e  =  <. C ,  D >. )
1716fveq2d 5529 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  ( 2nd `  <. C ,  D >. ) )
18 op2ndg 6133 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( 2nd `  <. C ,  D >. )  =  D )
198, 9, 18syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  D >. )  =  D )
2019ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  <. C ,  D >. )  =  D )
2117, 20eqtrd 2315 . . . . 5  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  D )
22 simplr 731 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  c  =  C )
2322fveq2d 5529 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  (
Base `  C )
)
24 curfval.a . . . . . . . 8  |-  A  =  ( Base `  C
)
2523, 24syl6eqr 2333 . . . . . . 7  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  A )
26 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  d  =  D )
2726fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  (
Base `  D )
)
28 curfval.b . . . . . . . . . 10  |-  B  =  ( Base `  D
)
2927, 28syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  B )
30 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  f  =  F )
3130ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  f  =  F )
3231fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3332oveqd 5875 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
( 1st `  f
) y )  =  ( x ( 1st `  F ) y ) )
3429, 33mpteq12dv 4098 . . . . . . . 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 5529 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  (  Hom  `  D )
)
36 curfval.j . . . . . . . . . . . 12  |-  J  =  (  Hom  `  D
)
3735, 36syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  J )
3837oveqd 5875 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y
(  Hom  `  d ) z )  =  ( y J z ) )
3931fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 2nd `  f )  =  ( 2nd `  F ) )
4039oveqd 5875 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
)  =  ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) )
4122fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  ( Id `  C ) )
42 curfval.1 . . . . . . . . . . . . 13  |-  .1.  =  ( Id `  C )
4341, 42syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  .1.  )
4443fveq1d 5527 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  c ) `  x )  =  (  .1.  `  x )
)
45 eqidd 2284 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  g  =  g )
4640, 44, 45oveq123d 5879 . . . . . . . . . 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 4098 . . . . . . . . 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 5910 . . . . . . . 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 3804 . . . . . . 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 4098 . . . . . 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 5529 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  (  Hom  `  C )
)
52 curfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
5351, 52syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  H )
5453oveqd 5875 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
(  Hom  `  c ) y )  =  ( x H y ) )
5539oveqd 5875 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  z >. ( 2nd `  f ) <.
y ,  z >.
)  =  ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) )
5626fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  ( Id `  D ) )
57 curfval.i . . . . . . . . . . . 12  |-  I  =  ( Id `  D
)
5856, 57syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  I )
5958fveq1d 5527 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  d ) `  z )  =  ( I `  z ) )
6055, 45, 59oveq123d 5879 . . . . . . . . 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 4098 . . . . . . . 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 4098 . . . . . . 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 5910 . . . . . 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 3804 . . . . 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 3124 . . . 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 3124 . . 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 4237 . . . 4  |-  <. C ,  D >.  e.  _V
6867a1i 10 . . 3  |-  ( ph  -> 
<. C ,  D >.  e. 
_V )
69 curfval.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
70 elex 2796 . . . 4  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  F  e.  _V )
7169, 70syl 15 . . 3  |-  ( ph  ->  F  e.  _V )
72 opex 4237 . . . 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 10 . . 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 5975 . 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 2327 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566   Idccid 13567    Func cfunc 13728    X.c cxpc 13942   curryF ccurf 13984
This theorem is referenced by:  curf1fval  13998  curf2  14003  curfcl  14006  curfpropd  14007  curfuncf  14012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-curf 13988
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