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Theorem uncfcurf 14013
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 2283 . . . . . . 7  |-  ( <" C D E "> uncurryF  G )  =  (
<" C D E "> uncurryF  G )
2 uncfcurf.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
32adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
4 uncfcurf.f . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
5 funcrcl 13737 . . . . . . . . . 10  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  ( ( C  X.c  D )  e.  Cat  /\  E  e.  Cat )
)
64, 5syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( C  X.c  D
)  e.  Cat  /\  E  e.  Cat )
)
76simprd 449 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
87adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  E  e.  Cat )
9 uncfcurf.g . . . . . . . . 9  |-  G  =  ( <. C ,  D >. curryF  F
)
10 eqid 2283 . . . . . . . . 9  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
11 uncfcurf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
129, 10, 11, 2, 4curfcl 14006 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
14 eqid 2283 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2283 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
16 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  C
) )
17 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
181, 3, 8, 13, 14, 15, 16, 17uncf1 14010 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( ( 1st `  ( ( 1st `  G ) `
 x ) ) `
 y ) )
1911adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  C  e.  Cat )
204adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
21 eqid 2283 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  =  ( ( 1st `  G
) `  x )
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 14000 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  (
( 1st `  G
) `  x )
) `  y )  =  ( x ( 1st `  F ) y ) )
2318, 22eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) )
2423ralrimivva 2635 . . . 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 2283 . . . . . . . 8  |-  ( C  X.c  D )  =  ( C  X.c  D )
2625, 14, 15xpcbas 13952 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
27 eqid 2283 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
28 relfunc 13736 . . . . . . . 8  |-  Rel  (
( C  X.c  D ) 
Func  E )
291, 2, 7, 12uncfcl 14009 . . . . . . . 8  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D ) 
Func  E ) )
30 1st2ndbr 6169 . . . . . . . 8  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D )  Func  E
) )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3128, 29, 30sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
3226, 27, 31funcf1 13740 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )
)
33 ffn 5389 . . . . . 6  |-  ( ( 1st `  ( <" C D E "> uncurryF  G ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  E )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) ) )
3432, 33syl 15 . . . . 5  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) ) )
35 1st2ndbr 6169 . . . . . . . 8  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3628, 4, 35sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3726, 27, 36funcf1 13740 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( (
Base `  C )  X.  ( Base `  D
) ) --> ( Base `  E ) )
38 ffn 5389 . . . . . 6  |-  ( ( 1st `  F ) : ( ( Base `  C )  X.  ( Base `  D ) ) --> ( Base `  E
)  ->  ( 1st `  F )  Fn  (
( Base `  C )  X.  ( Base `  D
) ) )
3937, 38syl 15 . . . . 5  |-  ( ph  ->  ( 1st `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  D
) ) )
40 eqfnov2 5951 . . . . 5  |-  ( ( ( 1st `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  ( 1st `  F )  Fn  ( ( Base `  C )  X.  ( Base `  D ) ) )  ->  ( ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
)  <->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) ( x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) ) )
4134, 39, 40syl2anc 642 . . . 4  |-  ( ph  ->  ( ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
)  <->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  D ) ( x ( 1st `  ( <" C D E "> uncurryF  G ) ) y )  =  ( x ( 1st `  F
) y ) ) )
4224, 41mpbird 223 . . 3  |-  ( ph  ->  ( 1st `  ( <" C D E "> uncurryF  G ) )  =  ( 1st `  F
) )
432ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  D  e.  Cat )
447ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  E  e.  Cat )
4512ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )
4616adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  x  e.  ( Base `  C )
)
4746adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  x  e.  (
Base `  C )
)
4817adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  y  e.  ( Base `  D )
)
4948adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  y  e.  (
Base `  D )
)
50 eqid 2283 . . . . . . . . . . 11  |-  (  Hom  `  C )  =  (  Hom  `  C )
51 eqid 2283 . . . . . . . . . . 11  |-  (  Hom  `  D )  =  (  Hom  `  D )
52 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  z  e.  ( Base `  C )
)
5352adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  z  e.  (
Base `  C )
)
54 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  w  e.  ( Base `  D )
)
5554adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  w  e.  (
Base `  D )
)
56 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  f  e.  ( x (  Hom  `  C
) z ) )
57 simprr 733 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  g  e.  ( y (  Hom  `  D
) w ) )
581, 43, 44, 45, 14, 15, 47, 49, 50, 51, 53, 55, 56, 57uncf2 14011 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( f (
<. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( ( ( ( x ( 2nd `  G
) z ) `  f ) `  w
) ( <. (
( 1st `  (
( 1st `  G
) `  x )
) `  y ) ,  ( ( 1st `  ( ( 1st `  G
) `  x )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  z )
) `  w )
) ( ( y ( 2nd `  (
( 1st `  G
) `  x )
) w ) `  g ) ) )
5911ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  C  e.  Cat )
604ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  F  e.  ( ( C  X.c  D ) 
Func  E ) )
619, 14, 59, 43, 60, 15, 47, 21, 49curf11 14000 . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. x ,  y >. )
6361, 62syl6eq 2331 . . . . . . . . . . . . . 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 14000 . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . . 15  |-  ( x ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. x ,  w >. )
6664, 65syl6eq 2331 . . . . . . . . . . . . . 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 3804 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  G ) `
 z )  =  ( ( 1st `  G
) `  z )
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 14000 . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . 14  |-  ( z ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. z ,  w >. )
7169, 70syl6eq 2331 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( Id
`  D )  =  ( Id `  D
)
74 eqid 2283 . . . . . . . . . . . . . 14  |-  ( ( x ( 2nd `  G
) z ) `  f )  =  ( ( x ( 2nd `  G ) z ) `
 f )
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 14004 . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . 13  |-  ( f ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) ( ( Id
`  D ) `  w ) )  =  ( ( <. x ,  w >. ( 2nd `  F
) <. z ,  w >. ) `  <. f ,  ( ( Id
`  D ) `  w ) >. )
7775, 76syl6eq 2331 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( Id
`  C )  =  ( Id `  C
)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 14001 . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . 13  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  w >. ) g )  =  ( ( <. x ,  y >. ( 2nd `  F ) <.
x ,  w >. ) `
 <. ( ( Id
`  C ) `  x ) ,  g
>. )
8179, 80syl6eq 2331 . . . . . . . . . . . 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 5879 . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
84 eqid 2283 . . . . . . . . . . . 12  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
85 eqid 2283 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
8636ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
8786adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
88 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
8988ad2antlr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  <. x ,  y >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9089adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. x ,  y
>.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
91 opelxpi 4721 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. x ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9247, 55, 91syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. x ,  w >.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
93 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( Base `  C )  /\  w  e.  ( Base `  D
) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9493adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  <. z ,  w >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
9594adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. z ,  w >.  e.  ( ( Base `  C )  X.  ( Base `  D ) ) )
9614, 50, 78, 59, 47catidcl 13584 . . . . . . . . . . . . . 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 4721 . . . . . . . . . . . . . 14  |-  ( ( ( ( Id `  C ) `  x
)  e.  ( x (  Hom  `  C
) x )  /\  g  e.  ( y
(  Hom  `  D ) w ) )  ->  <. ( ( Id `  C ) `  x
) ,  g >.  e.  ( ( x (  Hom  `  C )
x )  X.  (
y (  Hom  `  D
) w ) ) )
9896, 57, 97syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  x ) ,  g
>.  e.  ( ( x (  Hom  `  C
) x )  X.  ( y (  Hom  `  D ) w ) ) )
9925, 14, 15, 50, 51, 47, 49, 47, 55, 83xpchom2 13960 . . . . . . . . . . . . 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 2360 . . . . . . . . . . . 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 13584 . . . . . . . . . . . . . 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 4721 . . . . . . . . . . . . . 14  |-  ( ( f  e.  ( x (  Hom  `  C
) z )  /\  ( ( Id `  D ) `  w
)  e.  ( w (  Hom  `  D
) w ) )  ->  <. f ,  ( ( Id `  D
) `  w ) >.  e.  ( ( x (  Hom  `  C
) z )  X.  ( w (  Hom  `  D ) w ) ) )
10356, 101, 102syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D ) ) )  /\  ( z  e.  ( Base `  C
)  /\  w  e.  ( Base `  D )
) )  /\  (
f  e.  ( x (  Hom  `  C
) z )  /\  g  e.  ( y
(  Hom  `  D ) w ) ) )  ->  <. f ,  ( ( Id `  D
) `  w ) >.  e.  ( ( x (  Hom  `  C
) z )  X.  ( w (  Hom  `  D ) w ) ) )
10425, 14, 15, 50, 51, 47, 55, 53, 55, 83xpchom2 13960 . . . . . . . . . . . . 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 2360 . . . . . . . . . . . 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 13745 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
108 eqid 2283 . . . . . . . . . . . . . . 15  |-  (comp `  D )  =  (comp `  D )
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 13961 . . . . . . . . . . . . . 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 5529 . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . 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 2333 . . . . . . . . . . . 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 13586 . . . . . . . . . . . . 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 13585 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . 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 2315 . . . . . . . . . . 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 2321 . . . . . . . . . 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 2315 . . . . . . . . 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 2635 . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  (  Hom  `  E )  =  (  Hom  `  E )
12131ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( 1st `  ( <" C D E "> uncurryF  G ) ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  ( <" C D E "> uncurryF  G ) ) )
12226, 83, 120, 121, 89, 94funcf2 13742 . . . . . . . . . . 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 13960 . . . . . . . . . . . 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 5380 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. z ,  w >. ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) )  <->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) ) ) )
125122, 124mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) ) )
126 ffn 5389 . . . . . . . . . 10  |-  ( (
<. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. x ,  y >.
) (  Hom  `  E
) ( ( 1st `  ( <" C D E "> uncurryF  G ) ) `  <. z ,  w >. ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
127125, 126syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
12826, 83, 120, 86, 89, 94funcf2 13742 . . . . . . . . . . 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 5380 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  F
) <. z ,  w >. ) : ( <.
x ,  y >.
(  Hom  `  ( C  X.c  D ) ) <.
z ,  w >. ) --> ( ( ( 1st `  F ) `  <. x ,  y >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. z ,  w >. ) )  <->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
) ) )
130128, 129mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
) )
131 ffn 5389 . . . . . . . . . 10  |-  ( (
<. x ,  y >.
( 2nd `  F
) <. z ,  w >. ) : ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) --> ( ( ( 1st `  F ) `
 <. x ,  y
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. z ,  w >. )
)  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
132130, 131syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )
133 eqfnov2 5951 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) )  /\  ( <.
x ,  y >.
( 2nd `  F
) <. z ,  w >. )  Fn  ( ( x (  Hom  `  C
) z )  X.  ( y (  Hom  `  D ) w ) ) )  ->  (
( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  <->  A. f  e.  (
x (  Hom  `  C
) z ) A. g  e.  ( y
(  Hom  `  D ) w ) ( f ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) ) )
134127, 132, 133syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( ( <. x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. )  <->  A. f  e.  (
x (  Hom  `  C
) z ) A. g  e.  ( y
(  Hom  `  D ) w ) ( f ( <. x ,  y
>. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) g )  =  ( f ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) g ) ) )
135119, 134mpbird 223 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) ) )  /\  ( z  e.  (
Base `  C )  /\  w  e.  ( Base `  D ) ) )  ->  ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
136135ralrimivva 2635 . . . . . 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 2635 . . . . 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 5866 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. ) )
139 oveq2 5866 . . . . . . . . 9  |-  ( v  =  <. z ,  w >.  ->  ( u ( 2nd `  F ) v )  =  ( u ( 2nd `  F
) <. z ,  w >. ) )
140138, 139eqeq12d 2297 . . . . . . . 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 4827 . . . . . . 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 5865 . . . . . . . . 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 5865 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( u ( 2nd `  F )
<. z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
144142, 143eqeq12d 2297 . . . . . . . 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 2585 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( u ( 2nd `  F )
<. z ,  w >. )  <->  A. z  e.  ( Base `  C ) A. w  e.  ( Base `  D ) ( <.
x ,  y >.
( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ) )
146141, 145syl5bb 248 . . . . . 6  |-  ( u  =  <. x ,  y
>.  ->  ( A. v  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) ) )
147146ralxp 4827 . . . . 5  |-  ( A. u  e.  ( ( Base `  C )  X.  ( Base `  D
) ) A. v  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  D
) A. z  e.  ( Base `  C
) A. w  e.  ( Base `  D
) ( <. x ,  y >. ( 2nd `  ( <" C D E "> uncurryF  G ) ) <.
z ,  w >. )  =  ( <. x ,  y >. ( 2nd `  F ) <.
z ,  w >. ) )
148137, 147sylibr 203 . . . 4  |-  ( ph  ->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) )
14926, 31funcfn2 13743 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) ) )
15026, 36funcfn2 13743 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
151 eqfnov2 5951 . . . . 5  |-  ( ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  Fn  ( ( ( Base `  C )  X.  ( Base `  D ) )  X.  ( ( Base `  C )  X.  ( Base `  D ) ) )  /\  ( 2nd `  F )  Fn  (
( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )  ->  ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
)  <->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) ) )
152149, 150, 151syl2anc 642 . . . 4  |-  ( ph  ->  ( ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
)  <->  A. u  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) A. v  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ( u ( 2nd `  ( <" C D E "> uncurryF  G ) ) v )  =  ( u ( 2nd `  F
) v ) ) )
153148, 152mpbird 223 . . 3  |-  ( ph  ->  ( 2nd `  ( <" C D E "> uncurryF  G ) )  =  ( 2nd `  F
) )
15442, 153opeq12d 3804 . 2  |-  ( ph  -> 
<. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
155 1st2nd 6166 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  ( <" C D E "> uncurryF  G )  e.  ( ( C  X.c  D )  Func  E
) )  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
15628, 29, 155sylancr 644 . 2  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  <. ( 1st `  ( <" C D E "> uncurryF  G ) ) ,  ( 2nd `  ( <" C D E "> uncurryF  G ) ) >.
)
157 1st2nd 6166 . . 3  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
15828, 4, 157sylancr 644 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
159154, 156, 1583eqtr4d 2325 1  |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023    X. cxp 4687   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   <"cs3 11492   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728   FuncCat cfuc 13816    X.c cxpc 13942   curryF ccurf 13984   uncurryF cuncf 13985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-s2 11498  df-s3 11499  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-func 13732  df-cofu 13734  df-nat 13817  df-fuc 13818  df-xpc 13946  df-1stf 13947  df-2ndf 13948  df-prf 13949  df-evlf 13987  df-curf 13988  df-uncf 13989
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