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Theorem 2ndfcl 14300
Description: The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfcl.t  |-  T  =  ( C  X.c  D )
1stfcl.c  |-  ( ph  ->  C  e.  Cat )
1stfcl.d  |-  ( ph  ->  D  e.  Cat )
2ndfcl.p  |-  Q  =  ( C  2ndF  D )
Assertion
Ref Expression
2ndfcl  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )

Proof of Theorem 2ndfcl
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfcl.t . . . 4  |-  T  =  ( C  X.c  D )
2 eqid 2438 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2438 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 14280 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2438 . . . 4  |-  (  Hom  `  T )  =  (  Hom  `  T )
6 1stfcl.c . . . 4  |-  ( ph  ->  C  e.  Cat )
7 1stfcl.d . . . 4  |-  ( ph  ->  D  e.  Cat )
8 2ndfcl.p . . . 4  |-  Q  =  ( C  2ndF  D )
91, 4, 5, 6, 7, 82ndfval 14296 . . 3  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )
>. )
10 fo2nd 6370 . . . . . . . 8  |-  2nd : _V -onto-> _V
11 fofun 5657 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  2nd
13 fvex 5745 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5745 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 4993 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 5960 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 655 . . . . . 6  |-  ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6428 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6361 . . . . 5  |-  ( Q  =  <. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 2nd  |`  ( x (  Hom  `  T ) y ) ) ) >.  ->  ( 2nd `  Q )  =  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 2nd  |`  ( x (  Hom  `  T ) y ) ) ) )
209, 19syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  Q
)  =  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) ) )
2120opeq2d 3993 . . 3  |-  ( ph  -> 
<. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  Q
) >.  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )
>. )
229, 21eqtr4d 2473 . 2  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  Q ) >. )
23 eqid 2438 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
24 eqid 2438 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2438 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
26 eqid 2438 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2438 . . . 4  |-  (comp `  D )  =  (comp `  D )
281, 6, 7xpccat 14292 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f2ndres 6372 . . . . 5  |-  ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) : ( ( Base `  C )  X.  ( Base `  D ) ) --> ( Base `  D
)
3029a1i 11 . . . 4  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  D )
)
31 eqid 2438 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )  =  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 2nd  |`  ( x (  Hom  `  T ) y ) ) )
32 ovex 6109 . . . . . . 7  |-  ( x (  Hom  `  T
) y )  e. 
_V
33 resfunexg 5960 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
x (  Hom  `  T
) y )  e. 
_V )  ->  ( 2nd  |`  ( x (  Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 655 . . . . . 6  |-  ( 2nd  |`  ( x (  Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6423 . . . . 5  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )  Fn  ( ( (
Base `  C )  X.  ( Base `  D
) )  X.  (
( Base `  C )  X.  ( Base `  D
) ) )
3620fneq1d 5539 . . . . 5  |-  ( ph  ->  ( ( 2nd `  Q
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  <-> 
( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 2nd  |`  ( x (  Hom  `  T ) y ) ) )  Fn  (
( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) ) )
3735, 36mpbiri 226 . . . 4  |-  ( ph  ->  ( 2nd `  Q
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f2ndres 6372 . . . . . 6  |-  ( 2nd  |`  ( ( ( 1st `  x ) (  Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) ) : ( ( ( 1st `  x ) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) )
396adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
42 simprr 735 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
431, 4, 5, 39, 40, 8, 41, 422ndf2 14298 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  Q ) y )  =  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )
44 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
451, 4, 44, 23, 5, 41, 42xpchom 14282 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x (  Hom  `  T )
y )  =  ( ( ( 1st `  x
) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) )
4645reseq2d 5149 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( 2nd  |`  (
x (  Hom  `  T
) y ) )  =  ( 2nd  |`  (
( ( 1st `  x
) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) ) )
4743, 46eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  Q ) y )  =  ( 2nd  |`  ( (
( 1st `  x
) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) ) )
4847feq1d 5583 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( x ( 2nd `  Q
) y ) : ( ( ( 1st `  x ) (  Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) )  <->  ( 2nd  |`  ( ( ( 1st `  x ) (  Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) ) : ( ( ( 1st `  x ) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) )
4938, 48mpbiri 226 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  Q ) y ) : ( ( ( 1st `  x
) (  Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) )
50 fvres 5748 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
5150ad2antrl 710 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
52 fvres 5748 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 2nd `  y
) )
5352ad2antll 711 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 2nd `  y
) )
5451, 53oveq12d 6102 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  x ) (  Hom  `  D ) ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  y ) )  =  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) )
5545, 54feq23d 5591 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( x ( 2nd `  Q
) y ) : ( x (  Hom  `  T ) y ) --> ( ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x ) (  Hom  `  D )
( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) )  <->  ( x
( 2nd `  Q
) y ) : ( ( ( 1st `  x ) (  Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) ) )
5649, 55mpbird 225 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  Q ) y ) : ( x (  Hom  `  T
) y ) --> ( ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) (  Hom  `  D ) ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  y ) ) )
5728adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 13912 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x (  Hom  `  T
) x ) )
60 fvres 5748 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  e.  ( x (  Hom  `  T ) x )  ->  ( ( 2nd  |`  ( x (  Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 2nd `  (
( Id `  T
) `  x )
) )
6159, 60syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 2nd  |`  ( x (  Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 2nd `  (
( Id `  T
) `  x )
) )
62 1st2nd2 6389 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5735 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2438 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
68 xp1st 6379 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6380 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 67, 25, 24, 69, 71xpcid 14291 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )  =  <. ( ( Id
`  C ) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `
 ( 2nd `  x
) ) >. )
7364, 72eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5745 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5745 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op2ndd 6361 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  =  <. ( ( Id
`  C ) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `
 ( 2nd `  x
) ) >.  ->  ( 2nd `  ( ( Id
`  T ) `  x ) )  =  ( ( Id `  D ) `  ( 2nd `  x ) ) )
7773, 76syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  (
( Id `  T
) `  x )
)  =  ( ( Id `  D ) `
 ( 2nd `  x
) ) )
7861, 77eqtrd 2470 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 2nd  |`  ( x (  Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( ( Id
`  D ) `  ( 2nd `  x ) ) )
791, 4, 5, 65, 66, 8, 58, 582ndf2 14298 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  Q ) x )  =  ( 2nd  |`  ( x
(  Hom  `  T ) x ) ) )
8079fveq1d 5733 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( x ( 2nd `  Q
) x ) `  ( ( Id `  T ) `  x
) )  =  ( ( 2nd  |`  (
x (  Hom  `  T
) x ) ) `
 ( ( Id
`  T ) `  x ) ) )
8150adantl 454 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
8281fveq2d 5735 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  D ) `  ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) )  =  ( ( Id `  D ) `  ( 2nd `  x ) ) )
8378, 80, 823eqtr4d 2480 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( x ( 2nd `  Q
) x ) `  ( ( Id `  T ) `  x
) )  =  ( ( Id `  D
) `  ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  x ) ) )
84283ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  T  e.  Cat )
85 simp21 991 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
86 simp22 992 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
87 simp23 993 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
88 simp3l 986 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  f  e.  ( x (  Hom  `  T ) y ) )
89 simp3r 987 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  g  e.  ( y (  Hom  `  T ) z ) )
904, 5, 26, 84, 85, 86, 87, 88, 89catcocl 13915 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  T )
z ) f )  e.  ( x (  Hom  `  T )
z ) )
91 fvres 5748 . . . . . . 7  |-  ( ( g ( <. x ,  y >. (comp `  T ) z ) f )  e.  ( x (  Hom  `  T
) z )  -> 
( ( 2nd  |`  (
x (  Hom  `  T
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  T )
z ) f ) )  =  ( 2nd `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
9290, 91syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( x (  Hom  `  T )
z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( 2nd `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
931, 4, 5, 26, 85, 86, 87, 88, 89, 27xpcco2nd 14287 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( 2nd `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 2nd `  g ) ( <. ( 2nd `  x
) ,  ( 2nd `  y ) >. (comp `  D ) ( 2nd `  z ) ) ( 2nd `  f ) ) )
9492, 93eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( x (  Hom  `  T )
z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 2nd `  g ) ( <. ( 2nd `  x
) ,  ( 2nd `  y ) >. (comp `  D ) ( 2nd `  z ) ) ( 2nd `  f ) ) )
9563ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  C  e.  Cat )
9673ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  D  e.  Cat )
971, 4, 5, 95, 96, 8, 85, 872ndf2 14298 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( x
( 2nd `  Q
) z )  =  ( 2nd  |`  (
x (  Hom  `  T
) z ) ) )
9897fveq1d 5733 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
x ( 2nd `  Q
) z ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 2nd  |`  ( x
(  Hom  `  T ) z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
9985, 50syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  x )  =  ( 2nd `  x
) )
10086, 52syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  y )  =  ( 2nd `  y
) )
10199, 100opeq12d 3994 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  <. ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >.  =  <. ( 2nd `  x ) ,  ( 2nd `  y
) >. )
102 fvres 5748 . . . . . . . 8  |-  ( z  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 z )  =  ( 2nd `  z
) )
10387, 102syl 16 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  z )  =  ( 2nd `  z
) )
104101, 103oveq12d 6102 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( <. ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  D ) ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) )  =  ( <. ( 2nd `  x
) ,  ( 2nd `  y ) >. (comp `  D ) ( 2nd `  z ) ) )
1051, 4, 5, 95, 96, 8, 86, 872ndf2 14298 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( y
( 2nd `  Q
) z )  =  ( 2nd  |`  (
y (  Hom  `  T
) z ) ) )
106105fveq1d 5733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
y ( 2nd `  Q
) z ) `  g )  =  ( ( 2nd  |`  (
y (  Hom  `  T
) z ) ) `
 g ) )
107 fvres 5748 . . . . . . . 8  |-  ( g  e.  ( y (  Hom  `  T )
z )  ->  (
( 2nd  |`  ( y (  Hom  `  T
) z ) ) `
 g )  =  ( 2nd `  g
) )
10889, 107syl 16 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( y (  Hom  `  T )
z ) ) `  g )  =  ( 2nd `  g ) )
109106, 108eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
y ( 2nd `  Q
) z ) `  g )  =  ( 2nd `  g ) )
1101, 4, 5, 95, 96, 8, 85, 862ndf2 14298 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( x
( 2nd `  Q
) y )  =  ( 2nd  |`  (
x (  Hom  `  T
) y ) ) )
111110fveq1d 5733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
x ( 2nd `  Q
) y ) `  f )  =  ( ( 2nd  |`  (
x (  Hom  `  T
) y ) ) `
 f ) )
112 fvres 5748 . . . . . . . 8  |-  ( f  e.  ( x (  Hom  `  T )
y )  ->  (
( 2nd  |`  ( x (  Hom  `  T
) y ) ) `
 f )  =  ( 2nd `  f
) )
11388, 112syl 16 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( ( 2nd  |`  ( x (  Hom  `  T )
y ) ) `  f )  =  ( 2nd `  f ) )
114111, 113eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
x ( 2nd `  Q
) y ) `  f )  =  ( 2nd `  f ) )
115104, 109, 114oveq123d 6105 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
( y ( 2nd `  Q ) z ) `
 g ) (
<. ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  D ) ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) ) ( ( x ( 2nd `  Q ) y ) `
 f ) )  =  ( ( 2nd `  g ) ( <.
( 2nd `  x
) ,  ( 2nd `  y ) >. (comp `  D ) ( 2nd `  z ) ) ( 2nd `  f ) ) )
11694, 98, 1153eqtr4d 2480 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  ( (
x ( 2nd `  Q
) z ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( ( y ( 2nd `  Q ) z ) `
 g ) (
<. ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  D ) ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) ) ( ( x ( 2nd `  Q ) y ) `
 f ) ) )
1174, 3, 5, 23, 24, 25, 26, 27, 28, 7, 30, 37, 56, 83, 116isfuncd 14067 . . 3  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
) )
118 df-br 4216 . . 3  |-  ( ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
)  <->  <. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  Q
) >.  e.  ( T 
Func  D ) )
119117, 118sylib 190 . 2  |-  ( ph  -> 
<. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  Q
) >.  e.  ( T 
Func  D ) )
12022, 119eqeltrd 2512 1  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4215    X. cxp 4879    |` cres 4883   Fun wfun 5451    Fn wfn 5452   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894   Idccid 13895    Func cfunc 14056    X.c cxpc 14270    2ndF c2ndf 14272
This theorem is referenced by:  prf2nd  14307  1st2ndprf  14308  uncfcl  14337  uncf1  14338  uncf2  14339  curf2ndf  14349  yonedalem1  14374  yonedalem21  14375  yonedalem22  14380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-func 14060  df-xpc 14274  df-2ndf 14276
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