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Theorem uncfcurf 15707
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 2454 . . . . . . 7  |-  ( <" C D E "> uncurryF  G )  =  (
<" C D E "> uncurryF  G )
2 uncfcurf.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
32adantr 463 . . . . . . 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 15351 . . . . . . . . . 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 461 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
87adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  E  e.  Cat )
9 uncfcurf.g . . . . . . . . 9  |-  G  =  ( <. C ,  D >. curryF  F
)
10 eqid 2454 . . . . . . . . 9  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
11 uncfcurf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
129, 10, 11, 2, 4curfcl 15700 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
1312adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
14 eqid 2454 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2454 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
16 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  C
) )
17 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
181, 3, 8, 13, 14, 15, 16, 17uncf1 15704 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( ( 1st `  ( ( 1st `  G ) `
 x ) ) `
 y ) )
1911adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  C  e.  Cat )
204adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
21 eqid 2454 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  =  ( ( 1st `  G
) `  x )
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 15694 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  (
( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
2318, 22eqtrd 2495 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
2423ralrimivva 2875 . . . 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 2454 . . . . . . . 8  |-  ( C  X.c  D )  =  ( C  X.c  D )
2625, 14, 15xpcbas 15646 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
27 eqid 2454 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
28 relfunc 15350 . . . . . . . 8  |-  Rel  (
( C  X.c  D ) 
Func  E )
291, 2, 7, 12uncfcl 15703 . . . . . . . 8  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D ) 
Func  E ) )
30 1st2ndbr 6822 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3226, 27, 31funcf1 15354 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )
)
33 ffn 5713 . . . . . 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 6822 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3726, 27, 36funcf1 15354 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( (
Base `  C )  X.  ( Base `  D
) ) --> ( Base `  E ) )
38 ffn 5713 . . . . . 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 6382 . . . . 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 659 . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 727 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  x  e.  ( Base `  C )
)
4746adantr 463 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  y  e.  ( Base `  D )
)
4948adantr 463 . . . . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
51 eqid 2454 . . . . . . . . . . 11  |-  ( Hom  `  D )  =  ( Hom  `  D )
52 simprl 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  z  e.  ( Base `  C )
)
5352adantr 463 . . . . . . . . . . 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 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  w  e.  ( Base `  D )
)
5554adantr 463 . . . . . . . . . . 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 754 . . . . . . . . . . 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 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 ) ) )  ->  g  e.  ( y ( Hom  `  D
) w ) )
581, 43, 44, 45, 14, 15, 47, 49, 50, 51, 53, 55, 56, 57uncf2 15705 . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 15694 . . . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. x ,  y >. )
6361, 62syl6eq 2511 . . . . . . . . . . . . . 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 15694 . . . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. x ,  w >. )
6664, 65syl6eq 2511 . . . . . . . . . . . . . 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 4211 . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  G ) `
 z )  =  ( ( 1st `  G
) `  z )
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 15694 . . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . . 14  |-  ( z ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. z ,  w >. )
7169, 70syl6eq 2511 . . . . . . . . . . . . 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 6288 . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( Id
`  D )  =  ( Id `  D
)
74 eqid 2454 . . . . . . . . . . . . . 14  |-  ( ( x ( 2nd `  G
) z ) `  f )  =  ( ( x ( 2nd `  G ) z ) `
 f )
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 15698 . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . 13  |-  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) )  =  ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
7775, 76syl6eq 2511 . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( Id
`  C )  =  ( Id `  C
)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 15695 . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . 13  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) g )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) `
 <. ( ( Id
`  C ) `  x ) ,  g
>. )
8179, 80syl6eq 2511 . . . . . . . . . . . 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 6291 . . . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
84 eqid 2454 . . . . . . . . . . . 12  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
85 eqid 2454 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
8636ad2antrr 723 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
8988ad2antlr 724 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 5020 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. x ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9247, 55, 91syl2anc 659 . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9493adantl 464 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 15171 . . . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 15654 . . . . . . . . . . . . 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 2545 . . . . . . . . . . . 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 15171 . . . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 15654 . . . . . . . . . . . . 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 2545 . . . . . . . . . . . 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 15359 . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
108 eqid 2454 . . . . . . . . . . . . . . 15  |-  (comp `  D )  =  (comp `  D )
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 15655 . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . 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 2513 . . . . . . . . . . . 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 15173 . . . . . . . . . . . . 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 15172 . . . . . . . . . . . . 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 6288 . . . . . . . . . . . 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 2495 . . . . . . . . . . 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 2501 . . . . . . . . . 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 2495 . . . . . . . . 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 2875 . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( Hom  `  E )  =  ( Hom  `  E )
12131ad2antrr 723 . . . . . . . . . . . 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 15356 . . . . . . . . . . 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 15654 . . . . . . . . . . . 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 5700 . . . . . . . . . . 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 5713 . . . . . . . . . 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 15356 . . . . . . . . . . 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 5700 . . . . . . . . . . 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 5713 . . . . . . . . . 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 6382 . . . . . . . . 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 659 . . . . . . . 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 2875 . . . . . 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 2875 . . . . 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 6278 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
139 oveq2 6278 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  F ) v )  =  ( u ( 2nd `  F
) <. z ,  w >. ) )
140138, 139eqeq12d 2476 . . . . . . . 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 5133 . . . . . . 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 6277 . . . . . . . . 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 6277 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  F )
<. z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
144142, 143eqeq12d 2476 . . . . . . . 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 2898 . . . . . . 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 5133 . . . . 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 15357 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) ) )
15026, 36funcfn2 15357 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
151 eqfnov2 6382 . . . . 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 659 . . . 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 4211 . 2  |-  ( ph  -> 
<. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
155 1st2nd 6819 . . 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 661 . 2  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
157 1st2nd 6819 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
15828, 4, 157sylancr 661 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
159154, 156, 1583eqtr4d 2505 1  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   <.cop 4022   class class class wbr 4439    X. cxp 4986   Rel wrel 4993    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   <"cs3 12798   Basecbs 14716   Hom chom 14795  compcco 14796   Catccat 15153   Idccid 15154    Func cfunc 15342   FuncCat cfuc 15430    X.c cxpc 15636   curryF ccurf 15678   uncurryF cuncf 15679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-hom 14808  df-cco 14809  df-cat 15157  df-cid 15158  df-func 15346  df-cofu 15348  df-nat 15431  df-fuc 15432  df-xpc 15640  df-1stf 15641  df-2ndf 15642  df-prf 15643  df-evlf 15681  df-curf 15682  df-uncf 15683
This theorem is referenced by: (None)
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