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Theorem 2ndfcl 14254
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 2408 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2408 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 14234 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2408 . . . 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 14250 . . 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 6330 . . . . . . . 8  |-  2nd : _V -onto-> _V
11 fofun 5617 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
1210, 11ax-mp 8 . . . . . . 7  |-  Fun  2nd
13 fvex 5705 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5705 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 4953 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 5920 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 654 . . . . . 6  |-  ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6388 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6321 . . . . 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 3955 . . 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 2443 . 2  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  Q ) >. )
23 eqid 2408 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
24 eqid 2408 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2408 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
26 eqid 2408 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2408 . . . 4  |-  (comp `  D )  =  (comp `  D )
281, 6, 7xpccat 14246 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f2ndres 6332 . . . . 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 2408 . . . . . 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 6069 . . . . . . 7  |-  ( x (  Hom  `  T
) y )  e. 
_V
33 resfunexg 5920 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
x (  Hom  `  T
) y )  e. 
_V )  ->  ( 2nd  |`  ( x (  Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 654 . . . . . 6  |-  ( 2nd  |`  ( x (  Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6383 . . . . 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 5499 . . . . 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 225 . . . 4  |-  ( ph  ->  ( 2nd `  Q
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f2ndres 6332 . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 733 . . . . . . . . 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 734 . . . . . . . . 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 14252 . . . . . . . 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 2408 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
451, 4, 44, 23, 5, 41, 42xpchom 14236 . . . . . . . . 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 5109 . . . . . . . 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 2440 . . . . . . 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 5543 . . . . . 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 225 . . . . 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 5708 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
5150ad2antrl 709 . . . . . . 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 5708 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 2nd `  y
) )
5352ad2antll 710 . . . . . . 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 6062 . . . . . 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 5551 . . . . 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 224 . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 13866 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x (  Hom  `  T
) x ) )
60 fvres 5708 . . . . . . 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 6349 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5695 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2408 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
68 xp1st 6339 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6340 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 67, 25, 24, 69, 71xpcid 14245 . . . . . . . 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 2440 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5705 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5705 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op2ndd 6321 . . . . . . 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 2440 . . . . 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 14252 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  Q ) x )  =  ( 2nd  |`  ( x
(  Hom  `  T ) x ) ) )
8079fveq1d 5693 . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
8281fveq2d 5695 . . . . 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 2450 . . . 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 978 . . . . . . . 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 990 . . . . . . . 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 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 ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
87 simp23 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 ) ) )  ->  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
88 simp3l 985 . . . . . . . 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 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 ) ) )  ->  g  e.  ( y (  Hom  `  T ) z ) )
904, 5, 26, 84, 85, 86, 87, 88, 89catcocl 13869 . . . . . . 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 5708 . . . . . . 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 14241 . . . . . 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 2440 . . . . 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 978 . . . . . . 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 978 . . . . . . 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 14252 . . . . . 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 5693 . . . . 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 3956 . . . . . . 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 5708 . . . . . . . 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 6062 . . . . . 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 14252 . . . . . . . 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 5693 . . . . . . 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 5708 . . . . . . . 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 2440 . . . . . 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 14252 . . . . . . . 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 5693 . . . . . . 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 5708 . . . . . . . 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 2440 . . . . . 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 6065 . . . . 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 2450 . . . 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 14021 . . 3  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
) )
118 df-br 4177 . . 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 189 . 2  |-  ( ph  -> 
<. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  Q
) >.  e.  ( T 
Func  D ) )
12022, 119eqeltrd 2482 1  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2920   <.cop 3781   class class class wbr 4176    X. cxp 4839    |` cres 4843   Fun wfun 5411    Fn wfn 5412   -->wf 5413   -onto->wfo 5415   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311   Basecbs 13428    Hom chom 13499  compcco 13500   Catccat 13848   Idccid 13849    Func cfunc 14010    X.c cxpc 14224    2ndF c2ndf 14226
This theorem is referenced by:  prf2nd  14261  1st2ndprf  14262  uncfcl  14291  uncf1  14292  uncf2  14293  curf2ndf  14303  yonedalem1  14328  yonedalem21  14329  yonedalem22  14334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-hom 13512  df-cco 13513  df-cat 13852  df-cid 13853  df-func 14014  df-xpc 14228  df-2ndf 14230
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