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Theorem 2ndfcl 15131
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 2454 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2454 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 15111 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2454 . . . 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 15127 . . 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 6710 . . . . . . . 8  |-  2nd : _V -onto-> _V
11 fofun 5732 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  2nd
13 fvex 5812 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5812 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 6621 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 6055 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 672 . . . . . 6  |-  ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6763 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
( Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6701 . . . . 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 4177 . . 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 2498 . 2  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  Q ) >. )
23 eqid 2454 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
24 eqid 2454 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2454 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
26 eqid 2454 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2454 . . . 4  |-  (comp `  D )  =  (comp `  D )
281, 6, 7xpccat 15123 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f2ndres 6712 . . . . 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 2454 . . . . . 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 6228 . . . . . . 7  |-  ( x ( Hom  `  T
) y )  e. 
_V
33 resfunexg 6055 . . . . . . 7  |-  ( ( Fun  2nd  /\  (
x ( Hom  `  T
) y )  e. 
_V )  ->  ( 2nd  |`  ( x ( Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 672 . . . . . 6  |-  ( 2nd  |`  ( x ( Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6756 . . . . 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 5612 . . . . 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 233 . . . 4  |-  ( ph  ->  ( 2nd `  Q
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f2ndres 6712 . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 755 . . . . . . . . 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 756 . . . . . . . . 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 15129 . . . . . . . 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 2454 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
451, 4, 44, 23, 5, 41, 42xpchom 15113 . . . . . . . . 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 5221 . . . . . . . 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 2495 . . . . . . 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 5657 . . . . . 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 233 . . . . 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 5816 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
5150ad2antrl 727 . . . . . . 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 5816 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 2nd `  y
) )
5352ad2antll 728 . . . . . . 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 6221 . . . . . 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 5665 . . . . 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 232 . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 14743 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x ( Hom  `  T
) x ) )
60 fvres 5816 . . . . . . 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 6726 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5806 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2454 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
68 xp1st 6719 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6720 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 67, 25, 24, 69, 71xpcid 15122 . . . . . . . 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 2495 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5812 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5812 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op2ndd 6701 . . . . . . 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 2495 . . . . 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 15129 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  Q ) x )  =  ( 2nd  |`  ( x
( Hom  `  T ) x ) ) )
8079fveq1d 5804 . . . . 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 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 2nd  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 2nd `  x
) )
8281fveq2d 5806 . . . . 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 2505 . . . 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 1009 . . . . . . . 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 1021 . . . . . . . 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 1022 . . . . . . . 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 1023 . . . . . . . 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 1016 . . . . . . . 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 1017 . . . . . . . 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 14746 . . . . . . 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 5816 . . . . . . 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 15118 . . . . . 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 2495 . . . . 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 1009 . . . . . . 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 1009 . . . . . . 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 15129 . . . . . 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 5804 . . . . 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 4178 . . . . . . 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 5816 . . . . . . . 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 6221 . . . . . 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 15129 . . . . . . . 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 5804 . . . . . . 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 5816 . . . . . . . 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 2495 . . . . . 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 15129 . . . . . . . 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 5804 . . . . . . 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 5816 . . . . . . . 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 2495 . . . . . 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 6224 . . . . 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 2505 . . . 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 14898 . . 3  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
) )
118 df-br 4404 . . 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 196 . 2  |-  ( ph  -> 
<. ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  Q
) >.  e.  ( T 
Func  D ) )
12022, 119eqeltrd 2542 1  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994   class class class wbr 4403    X. cxp 4949    |` cres 4953   Fun wfun 5523    Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   Basecbs 14296   Hom chom 14372  compcco 14373   Catccat 14725   Idccid 14726    Func cfunc 14887    X.c cxpc 15101    2ndF c2ndf 15103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-hom 14385  df-cco 14386  df-cat 14729  df-cid 14730  df-func 14891  df-xpc 15105  df-2ndf 15107
This theorem is referenced by:  prf2nd  15138  1st2ndprf  15139  uncfcl  15168  uncf1  15169  uncf2  15170  curf2ndf  15180  yonedalem1  15205  yonedalem21  15206  yonedalem22  15211
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