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Theorem uncfcurf 15045
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 2441 . . . . . . 7  |-  ( <" C D E "> uncurryF  G )  =  (
<" C D E "> uncurryF  G )
2 uncfcurf.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
32adantr 462 . . . . . . 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 14769 . . . . . . . . . 10  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  ( ( C  X.c  D )  e.  Cat  /\  E  e.  Cat )
)
64, 5syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( C  X.c  D
)  e.  Cat  /\  E  e.  Cat )
)
76simprd 460 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
87adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  E  e.  Cat )
9 uncfcurf.g . . . . . . . . 9  |-  G  =  ( <. C ,  D >. curryF  F
)
10 eqid 2441 . . . . . . . . 9  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
11 uncfcurf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
129, 10, 11, 2, 4curfcl 15038 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
1312adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
14 eqid 2441 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2441 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
16 simprl 750 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  C
) )
17 simprr 751 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
181, 3, 8, 13, 14, 15, 16, 17uncf1 15042 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( ( 1st `  ( ( 1st `  G ) `
 x ) ) `
 y ) )
1911adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  C  e.  Cat )
204adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
21 eqid 2441 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  =  ( ( 1st `  G
) `  x )
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 15032 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  (
( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
2318, 22eqtrd 2473 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
2423ralrimivva 2806 . . . 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 2441 . . . . . . . 8  |-  ( C  X.c  D )  =  ( C  X.c  D )
2625, 14, 15xpcbas 14984 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
27 eqid 2441 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
28 relfunc 14768 . . . . . . . 8  |-  Rel  (
( C  X.c  D ) 
Func  E )
291, 2, 7, 12uncfcl 15041 . . . . . . . 8  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D ) 
Func  E ) )
30 1st2ndbr 6622 . . . . . . . 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 658 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3226, 27, 31funcf1 14772 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )
)
33 ffn 5556 . . . . . 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 16 . . . . 5  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) ) )
35 1st2ndbr 6622 . . . . . . . 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 658 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3726, 27, 36funcf1 14772 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( (
Base `  C )  X.  ( Base `  D
) ) --> ( Base `  E ) )
38 ffn 5556 . . . . . 6  |-  ( ( 1st `  F ) : ( ( Base `  C )  X.  ( Base `  D ) ) --> ( Base `  E
)  ->  ( 1st `  F )  Fn  (
( Base `  C )  X.  ( Base `  D
) ) )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  ( 1st `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  D
) ) )
40 eqfnov2 6196 . . . . 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 656 . . . 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 232 . . 3  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
) )
432ad3antrrr 724 . . . . . . . . . . 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 724 . . . . . . . . . . 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 724 . . . . . . . . . . 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 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  x  e.  ( Base `  C )
)
4746adantr 462 . . . . . . . . . . 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 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  y  e.  ( Base `  D )
)
4948adantr 462 . . . . . . . . . . 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 2441 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
51 eqid 2441 . . . . . . . . . . 11  |-  ( Hom  `  D )  =  ( Hom  `  D )
52 simprl 750 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  z  e.  ( Base `  C )
)
5352adantr 462 . . . . . . . . . . 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 751 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  w  e.  ( Base `  D )
)
5554adantr 462 . . . . . . . . . . 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 750 . . . . . . . . . . 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 751 . . . . . . . . . . 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 15043 . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 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 15032 . . . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. x ,  y >. )
6361, 62syl6eq 2489 . . . . . . . . . . . . . 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 15032 . . . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. x ,  w >. )
6664, 65syl6eq 2489 . . . . . . . . . . . . . 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 4064 . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  G ) `
 z )  =  ( ( 1st `  G
) `  z )
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 15032 . . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . . 14  |-  ( z ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. z ,  w >. )
7169, 70syl6eq 2489 . . . . . . . . . . . . 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 6108 . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . 14  |-  ( Id
`  D )  =  ( Id `  D
)
74 eqid 2441 . . . . . . . . . . . . . 14  |-  ( ( x ( 2nd `  G
) z ) `  f )  =  ( ( x ( 2nd `  G ) z ) `
 f )
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 15036 . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . 13  |-  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) )  =  ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
7775, 76syl6eq 2489 . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . 14  |-  ( Id
`  C )  =  ( Id `  C
)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 15033 . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . 13  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) g )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) `
 <. ( ( Id
`  C ) `  x ) ,  g
>. )
8179, 80syl6eq 2489 . . . . . . . . . . . 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 6111 . . . . . . . . . . 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 2441 . . . . . . . . . . . 12  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
84 eqid 2441 . . . . . . . . . . . 12  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
85 eqid 2441 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
8636ad2antrr 720 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 4867 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
8988ad2antlr 721 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 4867 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. x ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9247, 55, 91syl2anc 656 . . . . . . . . . . . 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 4867 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9493adantl 463 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 14616 . . . . . . . . . . . . . 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 4867 . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . 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 14992 . . . . . . . . . . . . 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 2518 . . . . . . . . . . . 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 14616 . . . . . . . . . . . . . 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 4867 . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . 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 14992 . . . . . . . . . . . . 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 2518 . . . . . . . . . . . 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 14777 . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
108 eqid 2441 . . . . . . . . . . . . . . 15  |-  (comp `  D )  =  (comp `  D )
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 14993 . . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . 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 6093 . . . . . . . . . . . . 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 2491 . . . . . . . . . . . 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 14618 . . . . . . . . . . . . 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 14617 . . . . . . . . . . . . 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 6108 . . . . . . . . . . . 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 2473 . . . . . . . . . . 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 2479 . . . . . . . . . 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 2473 . . . . . . . . 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 2806 . . . . . . . 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 2441 . . . . . . . . . . . 12  |-  ( Hom  `  E )  =  ( Hom  `  E )
12131ad2antrr 720 . . . . . . . . . . . 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 14774 . . . . . . . . . . 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 14992 . . . . . . . . . . . 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 5544 . . . . . . . . . . 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 210 . . . . . . . . . 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 5556 . . . . . . . . . 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 16 . . . . . . . . 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 14774 . . . . . . . . . . 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 5544 . . . . . . . . . . 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 210 . . . . . . . . . 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 5556 . . . . . . . . . 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 16 . . . . . . . . 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 6196 . . . . . . . . 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 656 . . . . . . . 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 232 . . . . . . 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 2806 . . . . . 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 2806 . . . . 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 6098 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
139 oveq2 6098 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  F ) v )  =  ( u ( 2nd `  F
) <. z ,  w >. ) )
140138, 139eqeq12d 2455 . . . . . . . 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 4977 . . . . . . 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 6097 . . . . . . . . 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 6097 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  F )
<. z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
144142, 143eqeq12d 2455 . . . . . . . 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 2755 . . . . . . 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 257 . . . . . 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 4977 . . . . 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 212 . . . 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 14775 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) ) )
15026, 36funcfn2 14775 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
151 eqfnov2 6196 . . . . 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 656 . . . 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 232 . . 3  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
) )
15442, 153opeq12d 4064 . 2  |-  ( ph  -> 
<. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
155 1st2nd 6619 . . 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 658 . 2  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
157 1st2nd 6619 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
15828, 4, 157sylancr 658 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
159154, 156, 1583eqtr4d 2483 1  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   <.cop 3880   class class class wbr 4289    X. cxp 4834   Rel wrel 4841    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   <"cs3 12465   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599    Func cfunc 14760   FuncCat cfuc 14848    X.c cxpc 14974   curryF ccurf 15016   uncurryF cuncf 15017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-s2 12471  df-s3 12472  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-func 14764  df-cofu 14766  df-nat 14849  df-fuc 14850  df-xpc 14978  df-1stf 14979  df-2ndf 14980  df-prf 14981  df-evlf 15019  df-curf 15020  df-uncf 15021
This theorem is referenced by: (None)
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