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Theorem uncfcurf 14029
Description: Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
uncfcurf.g  |-  G  =  ( <. C ,  D >. curryF  F
)
uncfcurf.c  |-  ( ph  ->  C  e.  Cat )
uncfcurf.d  |-  ( ph  ->  D  e.  Cat )
uncfcurf.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
Assertion
Ref Expression
uncfcurf  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )

Proof of Theorem uncfcurf
Dummy variables  f 
g  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . 7  |-  ( <" C D E "> uncurryF  G )  =  (
<" C D E "> uncurryF  G )
2 uncfcurf.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
32adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
4 uncfcurf.f . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
5 funcrcl 13753 . . . . . . . . . 10  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  ( ( C  X.c  D )  e.  Cat  /\  E  e.  Cat )
)
64, 5syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( C  X.c  D
)  e.  Cat  /\  E  e.  Cat )
)
76simprd 449 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
87adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  E  e.  Cat )
9 uncfcurf.g . . . . . . . . 9  |-  G  =  ( <. C ,  D >. curryF  F
)
10 eqid 2296 . . . . . . . . 9  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
11 uncfcurf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
129, 10, 11, 2, 4curfcl 14022 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
14 eqid 2296 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2296 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
16 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  C
) )
17 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
181, 3, 8, 13, 14, 15, 16, 17uncf1 14026 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( ( 1st `  ( ( 1st `  G ) `
 x ) ) `
 y ) )
1911adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  C  e.  Cat )
204adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
21 eqid 2296 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  =  ( ( 1st `  G
) `  x )
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 14016 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  (
( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
2318, 22eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
2423ralrimivva 2648 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) ( x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
25 eqid 2296 . . . . . . . 8  |-  ( C  X.c  D )  =  ( C  X.c  D )
2625, 14, 15xpcbas 13968 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
27 eqid 2296 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
28 relfunc 13752 . . . . . . . 8  |-  Rel  (
( C  X.c  D ) 
Func  E )
291, 2, 7, 12uncfcl 14025 . . . . . . . 8  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D ) 
Func  E ) )
30 1st2ndbr 6185 . . . . . . . 8  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D )  Func  E
) )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3128, 29, 30sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3226, 27, 31funcf1 13756 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )
)
33 ffn 5405 . . . . . 6  |-  ( ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) ) )
3432, 33syl 15 . . . . 5  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) ) )
35 1st2ndbr 6185 . . . . . . . 8  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3628, 4, 35sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3726, 27, 36funcf1 13756 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( (
Base `  C )  X.  ( Base `  D
) ) --> ( Base `  E ) )
38 ffn 5405 . . . . . 6  |-  ( ( 1st `  F ) : ( ( Base `  C )  X.  ( Base `  D ) ) --> ( Base `  E
)  ->  ( 1st `  F )  Fn  (
( Base `  C )  X.  ( Base `  D
) ) )
3937, 38syl 15 . . . . 5  |-  ( ph  ->  ( 1st `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  D
) ) )
40 eqfnov2 5967 . . . . 5  |-  ( ( ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  ( 1st `  F )  Fn  ( ( Base `  C )  X.  ( Base `  D ) ) )  ->  ( ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
)  <->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) ( x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) ) )
4134, 39, 40syl2anc 642 . . . 4  |-  ( ph  ->  ( ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
)  <->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) ( x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) ) )
4224, 41mpbird 223 . . 3  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
) )
432ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  D  e.  Cat )
447ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  E  e.  Cat )
4512ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
4616adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  x  e.  ( Base `  C )
)
4746adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  x  e.  (
Base `  C )
)
4817adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  y  e.  ( Base `  D )
)
4948adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  y  e.  (
Base `  D )
)
50 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  C )  =  (  Hom  `  C )
51 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  D )  =  (  Hom  `  D )
52 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  z  e.  ( Base `  C )
)
5352adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  z  e.  (
Base `  C )
)
54 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  w  e.  ( Base `  D )
)
5554adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  w  e.  (
Base `  D )
)
56 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  f  e.  ( x (  Hom  `  C
) z ) )
57 simprr 733 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  g  e.  ( y (  Hom  `  D
) w ) )
581, 43, 44, 45, 14, 15, 47, 49, 50, 51, 53, 55, 56, 57uncf2 14027 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( f (
<. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( ( ( ( x ( 2nd `  G
) z ) `  f ) `  w
) ( <. (
( 1st `  (
( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  z )
) `  w )
) ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g ) ) )
5911ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  C  e.  Cat )
604ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  F  e.  ( ( C  X.c  D ) 
Func  E ) )
619, 14, 59, 43, 60, 15, 47, 21, 49curf11 14016 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
62 df-ov 5877 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. x ,  y >. )
6361, 62syl6eq 2344 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  y )  =  ( ( 1st `  F ) `  <. x ,  y >. )
)
649, 14, 59, 43, 60, 15, 47, 21, 55curf11 14016 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w )  =  ( x ( 1st `  F ) w ) )
65 df-ov 5877 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. x ,  w >. )
6664, 65syl6eq 2344 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w )  =  ( ( 1st `  F ) `  <. x ,  w >. )
)
6763, 66opeq12d 3820 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. ( ( 1st `  ( ( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >.  =  <. ( ( 1st `  F ) `  <. x ,  y >. ) ,  ( ( 1st `  F ) `  <. x ,  w >. ) >. )
68 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  G ) `
 z )  =  ( ( 1st `  G
) `  z )
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 14016 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  z )
) `  w )  =  ( z ( 1st `  F ) w ) )
70 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( z ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. z ,  w >. )
7169, 70syl6eq 2344 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  ( ( 1st `  G
) `  z )
) `  w )  =  ( ( 1st `  F ) `  <. z ,  w >. )
)
7267, 71oveq12d 5892 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( <. (
( 1st `  (
( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  z )
) `  w )
)  =  ( <.
( ( 1st `  F
) `  <. x ,  y >. ) ,  ( ( 1st `  F
) `  <. x ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. z ,  w >. ) ) )
73 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Id
`  D )  =  ( Id `  D
)
74 eqid 2296 . . . . . . . . . . . . . 14  |-  ( ( x ( 2nd `  G
) z ) `  f )  =  ( ( x ( 2nd `  G ) z ) `
 f )
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 14020 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( ( x ( 2nd `  G
) z ) `  f ) `  w
)  =  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) ) )
76 df-ov 5877 . . . . . . . . . . . . 13  |-  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) )  =  ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
7775, 76syl6eq 2344 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( ( x ( 2nd `  G
) z ) `  f ) `  w
)  =  ( (
<. x ,  w >. ( 2nd `  F )
<. z ,  w >. ) `
 <. f ,  ( ( Id `  D
) `  w ) >. ) )
78 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Id
`  C )  =  ( Id `  C
)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 14017 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g )  =  ( ( ( Id `  C ) `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) g ) )
80 df-ov 5877 . . . . . . . . . . . . 13  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) g )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) `
 <. ( ( Id
`  C ) `  x ) ,  g
>. )
8179, 80syl6eq 2344 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g )  =  ( ( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) `  <. (
( Id `  C
) `  x ) ,  g >. )
)
8272, 77, 81oveq123d 5895 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( ( ( x ( 2nd `  G ) z ) `
 f ) `  w ) ( <.
( ( 1st `  (
( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  z )
) `  w )
) ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g ) )  =  ( ( ( <.
x ,  w >. ( 2nd `  F )
<. z ,  w >. ) `
 <. f ,  ( ( Id `  D
) `  w ) >. ) ( <. (
( 1st `  F
) `  <. x ,  y >. ) ,  ( ( 1st `  F
) `  <. x ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. z ,  w >. ) ) ( (
<. x ,  y >.
( 2nd `  F
) <. x ,  w >. ) `  <. (
( Id `  C
) `  x ) ,  g >. )
) )
83 eqid 2296 . . . . . . . . . . . 12  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
84 eqid 2296 . . . . . . . . . . . 12  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
85 eqid 2296 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
8636ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
8786adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
88 opelxpi 4737 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
8988ad2antlr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  <. x ,  y >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9089adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. x ,  y
>.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
91 opelxpi 4737 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. x ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9247, 55, 91syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. x ,  w >.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
93 opelxpi 4737 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9493adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9594adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. z ,  w >.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
9614, 50, 78, 59, 47catidcl 13600 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( Id
`  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
97 opelxpi 4737 . . . . . . . . . . . . . 14  |-  ( ( ( ( Id `  C ) `  x
)  e.  ( x (  Hom  `  C
) x )  /\  g  e.  ( y
(  Hom  `  D ) w ) )  ->  <. ( ( Id `  C ) `  x
) ,  g >.  e.  ( ( x (  Hom  `  C )
x )  X.  (
y (  Hom  `  D
) w ) ) )
9896, 57, 97syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  x ) ,  g
>.  e.  ( ( x (  Hom  `  C
) x )  X.  ( y (  Hom  `  D ) w ) ) )
9925, 14, 15, 50, 51, 47, 49, 47, 55, 83xpchom2 13976 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. x ,  w >. )  =  ( ( x (  Hom  `  C ) x )  X.  ( y (  Hom  `  D )
w ) ) )
10098, 99eleqtrrd 2373 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  x ) ,  g
>.  e.  ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. x ,  w >. ) )
10115, 51, 73, 43, 55catidcl 13600 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( Id
`  D ) `  w )  e.  ( w (  Hom  `  D
) w ) )
102 opelxpi 4737 . . . . . . . . . . . . . 14  |-  ( ( f  e.  ( x (  Hom  `  C
) z )  /\  ( ( Id `  D ) `  w
)  e.  ( w (  Hom  `  D
) w ) )  ->  <. f ,  ( ( Id `  D
) `  w ) >.  e.  ( ( x (  Hom  `  C
) z )  X.  ( w (  Hom  `  D ) w ) ) )
10356, 101, 102syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. f ,  ( ( Id `  D
) `  w ) >.  e.  ( ( x (  Hom  `  C
) z )  X.  ( w (  Hom  `  D ) w ) ) )
10425, 14, 15, 50, 51, 47, 55, 53, 55, 83xpchom2 13976 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( <. x ,  w >. (  Hom  `  ( C  X.c  D ) ) <.
z ,  w >. )  =  ( ( x (  Hom  `  C
) z )  X.  ( w (  Hom  `  D ) w ) ) )
105103, 104eleqtrrd 2373 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. f ,  ( ( Id `  D
) `  w ) >.  e.  ( <. x ,  w >. (  Hom  `  ( C  X.c  D ) ) <.
z ,  w >. ) )
10626, 83, 84, 85, 87, 90, 92, 95, 100, 105funcco 13761 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) `  ( <.
f ,  ( ( Id `  D ) `
 w ) >.
( <. <. x ,  y
>. ,  <. x ,  w >. >. (comp `  ( C  X.c  D ) ) <.
z ,  w >. )
<. ( ( Id `  C ) `  x
) ,  g >.
) )  =  ( ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
( <. ( ( 1st `  F ) `  <. x ,  y >. ) ,  ( ( 1st `  F ) `  <. x ,  w >. ) >. (comp `  E )
( ( 1st `  F
) `  <. z ,  w >. ) ) ( ( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) `  <. (
( Id `  C
) `  x ) ,  g >. )
) )
107 eqid 2296 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
108 eqid 2296 . . . . . . . . . . . . . . 15  |-  (comp `  D )  =  (comp `  D )
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 13977 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( <. f ,  ( ( Id
`  D ) `  w ) >. ( <. <. x ,  y
>. ,  <. x ,  w >. >. (comp `  ( C  X.c  D ) ) <.
z ,  w >. )
<. ( ( Id `  C ) `  x
) ,  g >.
)  =  <. (
f ( <. x ,  x >. (comp `  C
) z ) ( ( Id `  C
) `  x )
) ,  ( ( ( Id `  D
) `  w )
( <. y ,  w >. (comp `  D )
w ) g )
>. )
110109fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) `  ( <.
f ,  ( ( Id `  D ) `
 w ) >.
( <. <. x ,  y
>. ,  <. x ,  w >. >. (comp `  ( C  X.c  D ) ) <.
z ,  w >. )
<. ( ( Id `  C ) `  x
) ,  g >.
) )  =  ( ( <. x ,  y
>. ( 2nd `  F
) <. z ,  w >. ) `  <. (
f ( <. x ,  x >. (comp `  C
) z ) ( ( Id `  C
) `  x )
) ,  ( ( ( Id `  D
) `  w )
( <. y ,  w >. (comp `  D )
w ) g )
>. ) )
111 df-ov 5877 . . . . . . . . . . . . 13  |-  ( ( f ( <. x ,  x >. (comp `  C
) z ) ( ( Id `  C
) `  x )
) ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ( ( ( Id
`  D ) `  w ) ( <.
y ,  w >. (comp `  D ) w ) g ) )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) `
 <. ( f (
<. x ,  x >. (comp `  C ) z ) ( ( Id `  C ) `  x
) ) ,  ( ( ( Id `  D ) `  w
) ( <. y ,  w >. (comp `  D
) w ) g ) >. )
112110, 111syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) `  ( <.
f ,  ( ( Id `  D ) `
 w ) >.
( <. <. x ,  y
>. ,  <. x ,  w >. >. (comp `  ( C  X.c  D ) ) <.
z ,  w >. )
<. ( ( Id `  C ) `  x
) ,  g >.
) )  =  ( ( f ( <.
x ,  x >. (comp `  C ) z ) ( ( Id `  C ) `  x
) ) ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) ( ( ( Id `  D ) `
 w ) (
<. y ,  w >. (comp `  D ) w ) g ) ) )
11314, 50, 78, 59, 47, 107, 53, 56catrid 13602 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( f (
<. x ,  x >. (comp `  C ) z ) ( ( Id `  C ) `  x
) )  =  f )
11415, 51, 73, 43, 49, 108, 55, 57catlid 13601 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( ( Id `  D ) `
 w ) (
<. y ,  w >. (comp `  D ) w ) g )  =  g )
115113, 114oveq12d 5892 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( f ( <. x ,  x >. (comp `  C )
z ) ( ( Id `  C ) `
 x ) ) ( <. x ,  y
>. ( 2nd `  F
) <. z ,  w >. ) ( ( ( Id `  D ) `
 w ) (
<. y ,  w >. (comp `  D ) w ) g ) )  =  ( f ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) g ) )
116112, 115eqtrd 2328 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) `  ( <.
f ,  ( ( Id `  D ) `
 w ) >.
( <. <. x ,  y
>. ,  <. x ,  w >. >. (comp `  ( C  X.c  D ) ) <.
z ,  w >. )
<. ( ( Id `  C ) `  x
) ,  g >.
) )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) )
11782, 106, 1163eqtr2d 2334 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( ( ( ( x ( 2nd `  G ) z ) `
 f ) `  w ) ( <.
( ( 1st `  (
( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  z )
) `  w )
) ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g ) )  =  ( f ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. ) g ) )
11858, 117eqtrd 2328 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( f (
<. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) )
119118ralrimivva 2648 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  A. f  e.  ( x (  Hom  `  C ) z ) A. g  e.  ( y (  Hom  `  D
) w ) ( f ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) )
120 eqid 2296 . . . . . . . . . . . 12  |-  (  Hom  `  E )  =  (  Hom  `  E )
12131ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
12226, 83, 120, 121, 89, 94funcf2 13758 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. z ,  w >. ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) ) )
12325, 14, 15, 50, 51, 46, 48, 52, 54, 83xpchom2 13976 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. z ,  w >. )  =  ( ( x (  Hom  `  C ) z )  X.  ( y (  Hom  `  D )
w ) ) )
124123feq2d 5396 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. z ,  w >. ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) )  <->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) ) ) )
125122, 124mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) ) )
126 ffn 5405 . . . . . . . . . 10  |-  ( (
<. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
127125, 126syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
12826, 83, 120, 86, 89, 94funcf2 13758 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) : ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. z ,  w >. ) --> ( ( ( 1st `  F
) `  <. x ,  y >. ) (  Hom  `  E ) ( ( 1st `  F ) `
 <. z ,  w >. ) ) )
129123feq2d 5396 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  F
) <. z ,  w >. ) : ( <.
x ,  y >.
(  Hom  `  ( C  X.c  D ) ) <.
z ,  w >. ) --> ( ( ( 1st `  F ) `  <. x ,  y >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. z ,  w >. ) )  <->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
) ) )
130128, 129mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
) )
131 ffn 5405 . . . . . . . . . 10  |-  ( (
<. x ,  y >.
( 2nd `  F
) <. z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
)  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
132130, 131syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
133 eqfnov2 5967 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) )  /\  ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )  ->  (
( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  <->  A. f  e.  (
x (  Hom  `  C
) z ) A. g  e.  ( y
(  Hom  `  D ) w ) ( f ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) ) )
134127, 132, 133syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  <->  A. f  e.  (
x (  Hom  `  C
) z ) A. g  e.  ( y
(  Hom  `  D ) w ) ( f ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) ) )
135119, 134mpbird 223 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
136135ralrimivva 2648 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
137136ralrimivva 2648 . . . . 5  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) A. z  e.  ( Base `  C ) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
138 oveq2 5882 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
139 oveq2 5882 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  F ) v )  =  ( u ( 2nd `  F
) <. z ,  w >. ) )
140138, 139eqeq12d 2310 . . . . . . . 8  |-  ( v  =  <. z ,  w >.  ->  ( ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  ( u
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( u ( 2nd `  F )
<. z ,  w >. ) ) )
141140ralxp 4843 . . . . . . 7  |-  ( A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( u ( 2nd `  F )
<. z ,  w >. ) )
142 oveq1 5881 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
143 oveq1 5881 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  F )
<. z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
144142, 143eqeq12d 2310 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( u ( 2nd `  F )
<. z ,  w >. )  <-> 
( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ) )
1451442ralbidv 2598 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( u ( 2nd `  F )
<. z ,  w >. )  <->  A. z  e.  ( Base `  C ) A. w  e.  ( Base `  D ) ( <.
x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ) )
146141, 145syl5bb 248 . . . . . 6  |-  ( u  =  <. x ,  y
>.  ->  ( A. v  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ) )
147146ralxp 4843 . . . . 5  |-  ( A. u  e.  ( ( Base `  C )  X.  ( Base `  D
) ) A. v  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  D
) A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
148137, 147sylibr 203 . . . 4  |-  ( ph  ->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) )
14926, 31funcfn2 13759 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) ) )
15026, 36funcfn2 13759 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
151 eqfnov2 5967 . . . . 5  |-  ( ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) )  /\  ( 2nd `  F )  Fn  (
( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )  ->  ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
)  <->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) ) )
152149, 150, 151syl2anc 642 . . . 4  |-  ( ph  ->  ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
)  <->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) ) )
153148, 152mpbird 223 . . 3  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
) )
15442, 153opeq12d 3820 . 2  |-  ( ph  -> 
<. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
155 1st2nd 6182 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D )  Func  E
) )  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
15628, 29, 155sylancr 644 . 2  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
157 1st2nd 6182 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
15828, 4, 157sylancr 644 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
159154, 156, 1583eqtr4d 2338 1  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039    X. cxp 4703   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   <"cs3 11508   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583    Func cfunc 13744   FuncCat cfuc 13832    X.c cxpc 13958   curryF ccurf 14000   uncurryF cuncf 14001
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-s2 11514  df-s3 11515  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-func 13748  df-cofu 13750  df-nat 13833  df-fuc 13834  df-xpc 13962  df-1stf 13963  df-2ndf 13964  df-prf 13965  df-evlf 14003  df-curf 14004  df-uncf 14005
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