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Theorem uncfcurf 15366
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 2467 . . . . . . 7  |-  ( <" C D E "> uncurryF  G )  =  (
<" C D E "> uncurryF  G )
2 uncfcurf.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
32adantr 465 . . . . . . 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 15090 . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
87adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  E  e.  Cat )
9 uncfcurf.g . . . . . . . . 9  |-  G  =  ( <. C ,  D >. curryF  F
)
10 eqid 2467 . . . . . . . . 9  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
11 uncfcurf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
129, 10, 11, 2, 4curfcl 15359 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
1312adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
14 eqid 2467 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2467 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
16 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  C
) )
17 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
181, 3, 8, 13, 14, 15, 16, 17uncf1 15363 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( ( 1st `  ( ( 1st `  G ) `
 x ) ) `
 y ) )
1911adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  C  e.  Cat )
204adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
21 eqid 2467 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  =  ( ( 1st `  G
) `  x )
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 15353 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  (
( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
2318, 22eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
2423ralrimivva 2885 . . . 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 2467 . . . . . . . 8  |-  ( C  X.c  D )  =  ( C  X.c  D )
2625, 14, 15xpcbas 15305 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
27 eqid 2467 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
28 relfunc 15089 . . . . . . . 8  |-  Rel  (
( C  X.c  D ) 
Func  E )
291, 2, 7, 12uncfcl 15362 . . . . . . . 8  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D ) 
Func  E ) )
30 1st2ndbr 6833 . . . . . . . 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 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3226, 27, 31funcf1 15093 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )
)
33 ffn 5731 . . . . . 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 6833 . . . . . . . 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 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3726, 27, 36funcf1 15093 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( (
Base `  C )  X.  ( Base `  D
) ) --> ( Base `  E ) )
38 ffn 5731 . . . . . 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 6393 . . . . 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 661 . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 729 . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  x  e.  ( Base `  C )
)
4746adantr 465 . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  y  e.  ( Base `  D )
)
4948adantr 465 . . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
51 eqid 2467 . . . . . . . . . . 11  |-  ( Hom  `  D )  =  ( Hom  `  D )
52 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  z  e.  ( Base `  C )
)
5352adantr 465 . . . . . . . . . . 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 756 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  w  e.  ( Base `  D )
)
5554adantr 465 . . . . . . . . . . 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 755 . . . . . . . . . . 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 756 . . . . . . . . . . 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 15364 . . . . . . . . . 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 729 . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . 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 15353 . . . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. x ,  y >. )
6361, 62syl6eq 2524 . . . . . . . . . . . . . 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 15353 . . . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. x ,  w >. )
6664, 65syl6eq 2524 . . . . . . . . . . . . . 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 4221 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  G ) `
 z )  =  ( ( 1st `  G
) `  z )
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 15353 . . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . . 14  |-  ( z ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. z ,  w >. )
7169, 70syl6eq 2524 . . . . . . . . . . . . 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 6302 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( Id
`  D )  =  ( Id `  D
)
74 eqid 2467 . . . . . . . . . . . . . 14  |-  ( ( x ( 2nd `  G
) z ) `  f )  =  ( ( x ( 2nd `  G ) z ) `
 f )
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 15357 . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . 13  |-  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) )  =  ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
7775, 76syl6eq 2524 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( Id
`  C )  =  ( Id `  C
)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 15354 . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . 13  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) g )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) `
 <. ( ( Id
`  C ) `  x ) ,  g
>. )
8179, 80syl6eq 2524 . . . . . . . . . . . 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 6305 . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
84 eqid 2467 . . . . . . . . . . . 12  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
85 eqid 2467 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
8636ad2antrr 725 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
8988ad2antlr 726 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 5031 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. x ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9247, 55, 91syl2anc 661 . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9493adantl 466 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 14937 . . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 15313 . . . . . . . . . . . . 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 2558 . . . . . . . . . . . 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 14937 . . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 15313 . . . . . . . . . . . . 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 2558 . . . . . . . . . . . 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 15098 . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
108 eqid 2467 . . . . . . . . . . . . . . 15  |-  (comp `  D )  =  (comp `  D )
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 15314 . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . 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 2526 . . . . . . . . . . . 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 14939 . . . . . . . . . . . . 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 14938 . . . . . . . . . . . . 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 6302 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 2514 . . . . . . . . . 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 2508 . . . . . . . . 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 2885 . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( Hom  `  E )  =  ( Hom  `  E )
12131ad2antrr 725 . . . . . . . . . . . 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 15095 . . . . . . . . . . 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 15313 . . . . . . . . . . . 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 5718 . . . . . . . . . . 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 5731 . . . . . . . . . 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 15095 . . . . . . . . . . 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 5718 . . . . . . . . . . 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 5731 . . . . . . . . . 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 6393 . . . . . . . . 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 661 . . . . . . . 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 2885 . . . . . 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 2885 . . . . 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 6292 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
139 oveq2 6292 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  F ) v )  =  ( u ( 2nd `  F
) <. z ,  w >. ) )
140138, 139eqeq12d 2489 . . . . . . . 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 5144 . . . . . . 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 6291 . . . . . . . . 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 6291 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  F )
<. z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
144142, 143eqeq12d 2489 . . . . . . . 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 2908 . . . . . . 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 5144 . . . . 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 15096 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) ) )
15026, 36funcfn2 15096 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
151 eqfnov2 6393 . . . . 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 661 . . . 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 4221 . 2  |-  ( ph  -> 
<. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
155 1st2nd 6830 . . 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 663 . 2  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
157 1st2nd 6830 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
15828, 4, 157sylancr 663 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
159154, 156, 1583eqtr4d 2518 1  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447    X. cxp 4997   Rel wrel 5004    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   <"cs3 12770   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919   Idccid 14920    Func cfunc 15081   FuncCat cfuc 15169    X.c cxpc 15295   curryF ccurf 15337   uncurryF cuncf 15338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-s2 12776  df-s3 12777  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-hom 14579  df-cco 14580  df-cat 14923  df-cid 14924  df-func 15085  df-cofu 15087  df-nat 15170  df-fuc 15171  df-xpc 15299  df-1stf 15300  df-2ndf 15301  df-prf 15302  df-evlf 15340  df-curf 15341  df-uncf 15342
This theorem is referenced by: (None)
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