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Theorem 1stfcl 16082
Description: The first projection functor is a functor onto the left 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 )
1stfcl.p  |-  P  =  ( C  1stF  D )
Assertion
Ref Expression
1stfcl  |-  ( ph  ->  P  e.  ( T 
Func  C ) )

Proof of Theorem 1stfcl
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 2451 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2451 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 16063 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2451 . . . 4  |-  ( Hom  `  T )  =  ( Hom  `  T )
6 1stfcl.c . . . 4  |-  ( ph  ->  C  e.  Cat )
7 1stfcl.d . . . 4  |-  ( ph  ->  D  e.  Cat )
8 1stfcl.p . . . 4  |-  P  =  ( C  1stF  D )
91, 4, 5, 6, 7, 81stfval 16076 . . 3  |-  ( ph  ->  P  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )
>. )
10 fo1st 6813 . . . . . . . 8  |-  1st : _V -onto-> _V
11 fofun 5794 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  1st
13 fvex 5875 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5875 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 6595 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 6130 . . . . . . 7  |-  ( ( Fun  1st  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 678 . . . . . 6  |-  ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6870 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6804 . . . . 5  |-  ( P  =  <. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 1st  |`  ( x ( Hom  `  T ) y ) ) ) >.  ->  ( 2nd `  P )  =  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 1st  |`  ( x ( Hom  `  T ) y ) ) ) )
209, 19syl 17 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) ) )
2120opeq2d 4173 . . 3  |-  ( ph  -> 
<. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )
>. )
229, 21eqtr4d 2488 . 2  |-  ( ph  ->  P  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  P ) >. )
23 eqid 2451 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
24 eqid 2451 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2451 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
26 eqid 2451 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2451 . . . 4  |-  (comp `  C )  =  (comp `  C )
281, 6, 7xpccat 16075 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f1stres 6815 . . . . 5  |-  ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) : ( ( Base `  C )  X.  ( Base `  D ) ) --> ( Base `  C
)
3029a1i 11 . . . 4  |-  ( ph  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) : ( ( Base `  C
)  X.  ( Base `  D ) ) --> (
Base `  C )
)
31 eqid 2451 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )  =  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  D
) )  |->  ( 1st  |`  ( x ( Hom  `  T ) y ) ) )
32 ovex 6318 . . . . . . 7  |-  ( x ( Hom  `  T
) y )  e. 
_V
33 resfunexg 6130 . . . . . . 7  |-  ( ( Fun  1st  /\  (
x ( Hom  `  T
) y )  e. 
_V )  ->  ( 1st  |`  ( x ( Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 678 . . . . . 6  |-  ( 1st  |`  ( x ( Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6862 . . . . 5  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )  Fn  ( ( (
Base `  C )  X.  ( Base `  D
) )  X.  (
( Base `  C )  X.  ( Base `  D
) ) )
3620fneq1d 5666 . . . . 5  |-  ( ph  ->  ( ( 2nd `  P
)  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
) )  |->  ( 1st  |`  ( x ( Hom  `  T ) y ) ) )  Fn  (
( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) ) )
3735, 36mpbiri 237 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f1stres 6815 . . . . . 6  |-  ( 1st  |`  ( ( ( 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
) ) ) --> ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) )
396adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 764 . . . . . . . . 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 766 . . . . . . . . 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, 421stf2 16078 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  P ) y )  =  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )
44 eqid 2451 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
451, 4, 23, 44, 5, 41, 42xpchom 16065 . . . . . . . . 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 5105 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( 1st  |`  (
x ( Hom  `  T
) y ) )  =  ( 1st  |`  (
( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) ( Hom  `  D
) ( 2nd `  y
) ) ) ) )
4743, 46eqtrd 2485 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  P ) y )  =  ( 1st  |`  ( (
( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) ( Hom  `  D
) ( 2nd `  y
) ) ) ) )
4847feq1d 5714 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( x ( 2nd `  P
) y ) : ( ( ( 1st `  x ) ( Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) ( Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) )  <->  ( 1st  |`  ( ( ( 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
) ) ) --> ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) ) ) )
4938, 48mpbiri 237 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  P ) y ) : ( ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) )  X.  (
( 2nd `  x
) ( Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) ) )
50 fvres 5879 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
5150ad2antrl 734 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
52 fvres 5879 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 1st `  y
) )
5352ad2antll 735 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 1st `  y
) )
5451, 53oveq12d 6308 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ( Hom  `  C ) ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  y ) )  =  ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) ) )
5545, 54feq23d 5723 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( ( x ( 2nd `  P
) y ) : ( x ( Hom  `  T ) y ) --> ( ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x ) ( Hom  `  C )
( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) )  <->  ( x
( 2nd `  P
) y ) : ( ( ( 1st `  x ) ( Hom  `  C ) ( 1st `  y ) )  X.  ( ( 2nd `  x
) ( Hom  `  D
) ( 2nd `  y
) ) ) --> ( ( 1st `  x
) ( Hom  `  C
) ( 1st `  y
) ) ) )
5649, 55mpbird 236 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  ( x ( 2nd `  P ) y ) : ( x ( Hom  `  T
) y ) --> ( ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ( Hom  `  C ) ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  y ) ) )
5728adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 15588 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x ( Hom  `  T
) x ) )
60 fvres 5879 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  e.  ( x ( Hom  `  T ) x )  ->  ( ( 1st  |`  ( x ( Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 1st `  (
( Id `  T
) `  x )
) )
6159, 60syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( x ( Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 1st `  (
( Id `  T
) `  x )
) )
62 1st2nd2 6830 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5869 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2451 . . . . . . . . 9  |-  ( Id
`  D )  =  ( Id `  D
)
68 xp1st 6823 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6824 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 25, 67, 24, 69, 71xpcid 16074 . . . . . . . 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 2485 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5875 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5875 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op1std 6803 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  =  <. ( ( Id
`  C ) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `
 ( 2nd `  x
) ) >.  ->  ( 1st `  ( ( Id
`  T ) `  x ) )  =  ( ( Id `  C ) `  ( 1st `  x ) ) )
7773, 76syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  (
( Id `  T
) `  x )
)  =  ( ( Id `  C ) `
 ( 1st `  x
) ) )
7861, 77eqtrd 2485 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( x ( Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( ( Id
`  C ) `  ( 1st `  x ) ) )
791, 4, 5, 65, 66, 8, 58, 581stf2 16078 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  P ) x )  =  ( 1st  |`  ( x
( Hom  `  T ) x ) ) )
8079fveq1d 5867 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( x ( 2nd `  P
) x ) `  ( ( Id `  T ) `  x
) )  =  ( ( 1st  |`  (
x ( Hom  `  T
) x ) ) `
 ( ( Id
`  T ) `  x ) ) )
8150adantl 468 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
8281fveq2d 5869 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  C ) `  ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) )  =  ( ( Id `  C ) `  ( 1st `  x ) ) )
8378, 80, 823eqtr4d 2495 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( x ( 2nd `  P
) x ) `  ( ( Id `  T ) `  x
) )  =  ( ( Id `  C
) `  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  x ) ) )
84283ad2ant1 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 ) ) )  ->  T  e.  Cat )
85 simp21 1041 . . . . . . . 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 1042 . . . . . . . 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 1043 . . . . . . . 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 1036 . . . . . . . 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 1037 . . . . . . . 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 15591 . . . . . . 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 5879 . . . . . . 7  |-  ( ( g ( <. x ,  y >. (comp `  T ) z ) f )  e.  ( x ( Hom  `  T
) z )  -> 
( ( 1st  |`  (
x ( Hom  `  T
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  T )
z ) f ) )  =  ( 1st `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
9290, 91syl 17 . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( x ( Hom  `  T )
z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( 1st `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
931, 4, 5, 26, 85, 86, 87, 88, 89, 27xpcco1st 16069 . . . . . 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 ) ) )  ->  ( 1st `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 1st `  g ) ( <. ( 1st `  x
) ,  ( 1st `  y ) >. (comp `  C ) ( 1st `  z ) ) ( 1st `  f ) ) )
9492, 93eqtrd 2485 . . . . 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 ) ) )  ->  ( ( 1st  |`  ( x ( Hom  `  T )
z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 1st `  g ) ( <. ( 1st `  x
) ,  ( 1st `  y ) >. (comp `  C ) ( 1st `  z ) ) ( 1st `  f ) ) )
9563ad2ant1 1029 . . . . . . 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 1029 . . . . . . 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, 871stf2 16078 . . . . . 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 `  P
) z )  =  ( 1st  |`  (
x ( Hom  `  T
) z ) ) )
9897fveq1d 5867 . . . . 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 `  P
) z ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( 1st  |`  ( x
( Hom  `  T ) z ) ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) ) )
9985, 50syl 17 . . . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  x )  =  ( 1st `  x
) )
10086, 52syl 17 . . . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  y )  =  ( 1st `  y
) )
10199, 100opeq12d 4174 . . . . . . 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 ) ) )  ->  <. ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >.  =  <. ( 1st `  x ) ,  ( 1st `  y
) >. )
102 fvres 5879 . . . . . . . 8  |-  ( z  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 z )  =  ( 1st `  z
) )
10387, 102syl 17 . . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  z )  =  ( 1st `  z
) )
104101, 103oveq12d 6308 . . . . . 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 ) ) )  ->  ( <. ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  C ) ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) )  =  ( <. ( 1st `  x
) ,  ( 1st `  y ) >. (comp `  C ) ( 1st `  z ) ) )
1051, 4, 5, 95, 96, 8, 86, 871stf2 16078 . . . . . . . 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 `  P
) z )  =  ( 1st  |`  (
y ( Hom  `  T
) z ) ) )
106105fveq1d 5867 . . . . . . 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 `  P
) z ) `  g )  =  ( ( 1st  |`  (
y ( Hom  `  T
) z ) ) `
 g ) )
107 fvres 5879 . . . . . . . 8  |-  ( g  e.  ( y ( Hom  `  T )
z )  ->  (
( 1st  |`  ( y ( Hom  `  T
) z ) ) `
 g )  =  ( 1st `  g
) )
10889, 107syl 17 . . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( y ( Hom  `  T )
z ) ) `  g )  =  ( 1st `  g ) )
109106, 108eqtrd 2485 . . . . . 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 `  P
) z ) `  g )  =  ( 1st `  g ) )
1101, 4, 5, 95, 96, 8, 85, 861stf2 16078 . . . . . . . 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 `  P
) y )  =  ( 1st  |`  (
x ( Hom  `  T
) y ) ) )
111110fveq1d 5867 . . . . . . 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 `  P
) y ) `  f )  =  ( ( 1st  |`  (
x ( Hom  `  T
) y ) ) `
 f ) )
112 fvres 5879 . . . . . . . 8  |-  ( f  e.  ( x ( Hom  `  T )
y )  ->  (
( 1st  |`  ( x ( Hom  `  T
) y ) ) `
 f )  =  ( 1st `  f
) )
11388, 112syl 17 . . . . . . 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 ) ) )  ->  ( ( 1st  |`  ( x ( Hom  `  T )
y ) ) `  f )  =  ( 1st `  f ) )
114111, 113eqtrd 2485 . . . . . 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 `  P
) y ) `  f )  =  ( 1st `  f ) )
115104, 109, 114oveq123d 6311 . . . . 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 `  P ) z ) `
 g ) (
<. ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  C ) ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) ) ( ( x ( 2nd `  P ) y ) `
 f ) )  =  ( ( 1st `  g ) ( <.
( 1st `  x
) ,  ( 1st `  y ) >. (comp `  C ) ( 1st `  z ) ) ( 1st `  f ) ) )
11694, 98, 1153eqtr4d 2495 . . . 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 `  P
) z ) `  ( g ( <.
x ,  y >.
(comp `  T )
z ) f ) )  =  ( ( ( y ( 2nd `  P ) z ) `
 g ) (
<. ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  x ) ,  ( ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) `  y ) >. (comp `  C ) ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) `  z ) ) ( ( x ( 2nd `  P ) y ) `
 f ) ) )
1174, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 83, 116isfuncd 15770 . . 3  |-  ( ph  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  C )
( 2nd `  P
) )
118 df-br 4403 . . 3  |-  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  C )
( 2nd `  P
)  <->  <. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  e.  ( T 
Func  C ) )
119117, 118sylib 200 . 2  |-  ( ph  -> 
<. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  e.  ( T 
Func  C ) )
12022, 119eqeltrd 2529 1  |-  ( ph  ->  P  e.  ( T 
Func  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045   <.cop 3974   class class class wbr 4402    X. cxp 4832    |` cres 4836   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   Basecbs 15121   Hom chom 15201  compcco 15202   Catccat 15570   Idccid 15571    Func cfunc 15759    X.c cxpc 16053    1stF c1stf 16054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-hom 15214  df-cco 15215  df-cat 15574  df-cid 15575  df-func 15763  df-xpc 16057  df-1stf 16058
This theorem is referenced by:  prf1st  16089  1st2ndprf  16091  uncfcl  16120  uncf1  16121  uncf2  16122  diagcl  16126  diag11  16128  diag12  16129  diag2  16130  yonedalem1  16157  yonedalem21  16158  yonedalem22  16163
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