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Theorem 2ndfcl 15342
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 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2467 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 15322 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2467 . . . 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 15338 . . 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 6816 . . . . . . . 8  |-  2nd : _V -onto-> _V
11 fofun 5802 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  2nd
13 fvex 5882 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5882 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 6599 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 6137 . . . . . . 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 6872 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 2nd  |`  ( x
( Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6806 . . . . 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 4226 . . 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 2511 . 2  |-  ( ph  ->  Q  =  <. ( 2nd  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  Q ) >. )
23 eqid 2467 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
24 eqid 2467 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2467 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
26 eqid 2467 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2467 . . . 4  |-  (comp `  D )  =  (comp `  D )
281, 6, 7xpccat 15334 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f2ndres 6818 . . . . 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 2467 . . . . . 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 6320 . . . . . . 7  |-  ( x ( Hom  `  T
) y )  e. 
_V
33 resfunexg 6137 . . . . . . 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 6864 . . . . 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 5677 . . . . 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 6818 . . . . . 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 15340 . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
451, 4, 44, 23, 5, 41, 42xpchom 15324 . . . . . . . . 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 5279 . . . . . . . 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 2508 . . . . . . 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 5723 . . . . . 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 5886 . . . . . . . 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 5886 . . . . . . . 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 6313 . . . . . 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 5732 . . . . 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 14954 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x ( Hom  `  T
) x ) )
60 fvres 5886 . . . . . . 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 6832 . . . . . . . . . 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 5876 . . . . . . . 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 2467 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
68 xp1st 6825 . . . . . . . . . 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 6826 . . . . . . . . . 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 15333 . . . . . . . 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 2508 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5882 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5882 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op2ndd 6806 . . . . . . 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 2508 . . . . 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 15340 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  Q ) x )  =  ( 2nd  |`  ( x
( Hom  `  T ) x ) ) )
8079fveq1d 5874 . . . . 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 5876 . . . . 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 2518 . . . 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 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 ) ) )  ->  T  e.  Cat )
85 simp21 1029 . . . . . . . 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 1030 . . . . . . . 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 1031 . . . . . . . 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 1024 . . . . . . . 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 1025 . . . . . . . 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 14957 . . . . . . 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 5886 . . . . . . 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 15329 . . . . . 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 2508 . . . . 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 1017 . . . . . . 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 1017 . . . . . . 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 15340 . . . . . 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 5874 . . . . 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 4227 . . . . . . 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 5886 . . . . . . . 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 6313 . . . . . 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 15340 . . . . . . . 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 5874 . . . . . . 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 5886 . . . . . . . 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 2508 . . . . . 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 15340 . . . . . . . 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 5874 . . . . . . 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 5886 . . . . . . . 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 2508 . . . . . 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 6316 . . . . 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 2518 . . . 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 15109 . . 3  |-  ( ph  ->  ( 2nd  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  D )
( 2nd `  Q
) )
118 df-br 4454 . . 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 2555 1  |-  ( ph  ->  Q  e.  ( T 
Func  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039   class class class wbr 4453    X. cxp 5003    |` cres 5007   Fun wfun 5588    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   Basecbs 14507   Hom chom 14583  compcco 14584   Catccat 14936   Idccid 14937    Func cfunc 15098    X.c cxpc 15312    2ndF c2ndf 15314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-cat 14940  df-cid 14941  df-func 15102  df-xpc 15316  df-2ndf 15318
This theorem is referenced by:  prf2nd  15349  1st2ndprf  15350  uncfcl  15379  uncf1  15380  uncf2  15381  curf2ndf  15391  yonedalem1  15416  yonedalem21  15417  yonedalem22  15422
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