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Theorem curfval 14013
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 14004 . . . 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 5555 . . . . 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 5545 . . . . 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 6148 . . . . . . 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 2328 . . . 4  |-  ( (
ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  ->  ( 1st `  e )  =  C )
14 fvex 5555 . . . . . 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 5545 . . . . . 6  |-  ( ( ( ph  /\  (
e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  ->  ( 2nd `  e
)  =  ( 2nd `  <. C ,  D >. ) )
18 op2ndg 6149 . . . . . . . 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 2328 . . . . 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 5545 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  c )  =  (
Base `  C )
)
24 curfval.a . . . . . . . 8  |-  A  =  ( Base `  C
)
2523, 24syl6eqr 2346 . . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Base `  d )  =  (
Base `  D )
)
28 curfval.b . . . . . . . . . 10  |-  B  =  ( Base `  D
)
2927, 28syl6eqr 2346 . . . . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 1st `  f )  =  ( 1st `  F ) )
3332oveqd 5891 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
( 1st `  f
) y )  =  ( x ( 1st `  F ) y ) )
3429, 33mpteq12dv 4114 . . . . . . . 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 5545 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  (  Hom  `  D )
)
36 curfval.j . . . . . . . . . . . 12  |-  J  =  (  Hom  `  D
)
3735, 36syl6eqr 2346 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  d )  =  J )
3837oveqd 5891 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( y
(  Hom  `  d ) z )  =  ( y J z ) )
3931fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( 2nd `  f )  =  ( 2nd `  F ) )
4039oveqd 5891 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  y >. ( 2nd `  f ) <.
x ,  z >.
)  =  ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) )
4122fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  ( Id `  C ) )
42 curfval.1 . . . . . . . . . . . . 13  |-  .1.  =  ( Id `  C )
4341, 42syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  c )  =  .1.  )
4443fveq1d 5543 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  c ) `  x )  =  (  .1.  `  x )
)
45 eqidd 2297 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  g  =  g )
4640, 44, 45oveq123d 5895 . . . . . . . . . 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 4114 . . . . . . . . 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 5926 . . . . . . . 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 3820 . . . . . . 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 4114 . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  (  Hom  `  C )
)
52 curfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
5351, 52syl6eqr 2346 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  (  Hom  `  c )  =  H )
5453oveqd 5891 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( x
(  Hom  `  c ) y )  =  ( x H y ) )
5539oveqd 5891 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( <. x ,  z >. ( 2nd `  f ) <.
y ,  z >.
)  =  ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) )
5626fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  ( Id `  D ) )
57 curfval.i . . . . . . . . . . . 12  |-  I  =  ( Id `  D
)
5856, 57syl6eqr 2346 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( Id `  d )  =  I )
5958fveq1d 5543 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( e  =  <. C ,  D >.  /\  f  =  F ) )  /\  c  =  C )  /\  d  =  D
)  ->  ( ( Id `  d ) `  z )  =  ( I `  z ) )
6055, 45, 59oveq123d 5895 . . . . . . . . 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 4114 . . . . . . . 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 4114 . . . . . . 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 5926 . . . . . 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 3820 . . . . 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 3137 . . . 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 3137 . . 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 4253 . . . 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 2809 . . . 4  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  F  e.  _V )
7169, 70syl 15 . . 3  |-  ( ph  ->  F  e.  _V )
72 opex 4253 . . . 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 5991 . 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 2340 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 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   <.cop 3656    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582   Idccid 13583    Func cfunc 13744    X.c cxpc 13958   curryF ccurf 14000
This theorem is referenced by:  curf1fval  14014  curf2  14019  curfcl  14022  curfpropd  14023  curfuncf  14028
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-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-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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-curf 14004
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