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Theorem 2ndfcl 14285
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 2435 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2435 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 14265 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2435 . . . 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 14281 . . 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 6359 . . . . . . . 8  |-  2nd : _V -onto-> _V
11 fofun 5646 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
1210, 11ax-mp 8 . . . . . . 7  |-  Fun  2nd
13 fvex 5734 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5734 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 4982 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 5949 . . . . . . 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 6417 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
(  Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6350 . . . . 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 3983 . . 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 2470 . 2  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  Q ) >. )
23 eqid 2435 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
24 eqid 2435 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2435 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
26 eqid 2435 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2435 . . . 4  |-  (comp `  D )  =  (comp `  D )
281, 6, 7xpccat 14277 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f2ndres 6361 . . . . 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 2435 . . . . . 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 6098 . . . . . . 7  |-  ( x (  Hom  `  T
) y )  e. 
_V
33 resfunexg 5949 . . . . . . 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 6412 . . . . 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 5528 . . . . 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 6361 . . . . . 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 14283 . . . . . . . 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 2435 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
451, 4, 44, 23, 5, 41, 42xpchom 14267 . . . . . . . . 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 5138 . . . . . . . 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 2467 . . . . . . 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 5572 . . . . . 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 5737 . . . . . . . 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 5737 . . . . . . . 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 6091 . . . . . 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 5580 . . . . 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 13897 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x (  Hom  `  T
) x ) )
60 fvres 5737 . . . . . . 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 6378 . . . . . . . . . 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 5724 . . . . . . . 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 2435 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
68 xp1st 6368 . . . . . . . . . 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 6369 . . . . . . . . . 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 14276 . . . . . . . 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 2467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5734 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5734 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op2ndd 6350 . . . . . . 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 2467 . . . . 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 14283 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  Q ) x )  =  ( 2nd  |`  ( x
(  Hom  `  T ) x ) ) )
8079fveq1d 5722 . . . . 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 5724 . . . . 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 2477 . . . 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 13900 . . . . . . 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 5737 . . . . . . 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 14272 . . . . . 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 2467 . . . . 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 14283 . . . . . 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 5722 . . . . 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 3984 . . . . . . 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 5737 . . . . . . . 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 6091 . . . . . 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 14283 . . . . . . . 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 5722 . . . . . . 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 5737 . . . . . . . 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 2467 . . . . . 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 14283 . . . . . . . 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 5722 . . . . . . 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 5737 . . . . . . . 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 2467 . . . . . 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 6094 . . . . 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 2477 . . . 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 14052 . . 3  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
) )
118 df-br 4205 . . 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 2509 1  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204    X. cxp 4868    |` cres 4872   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13459    Hom chom 13530  compcco 13531   Catccat 13879   Idccid 13880    Func cfunc 14041    X.c cxpc 14255    2ndF c2ndf 14257
This theorem is referenced by:  prf2nd  14292  1st2ndprf  14293  uncfcl  14322  uncf1  14323  uncf2  14324  curf2ndf  14334  yonedalem1  14359  yonedalem21  14360  yonedalem22  14365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-hom 13543  df-cco 13544  df-cat 13883  df-cid 13884  df-func 14045  df-xpc 14259  df-2ndf 14261
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