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Theorem 1stfcl 15682
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 2402 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2402 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 15663 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2402 . . . 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 15676 . . 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 6758 . . . . . . . 8  |-  1st : _V -onto-> _V
11 fofun 5735 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  1st
13 fvex 5815 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5815 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 6542 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 6074 . . . . . . 7  |-  ( ( Fun  1st  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 670 . . . . . 6  |-  ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6815 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
( Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6749 . . . . 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 4165 . . 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 2446 . 2  |-  ( ph  ->  P  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  P ) >. )
23 eqid 2402 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
24 eqid 2402 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2402 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
26 eqid 2402 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2402 . . . 4  |-  (comp `  C )  =  (comp `  C )
281, 6, 7xpccat 15675 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f1stres 6760 . . . . 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 2402 . . . . . 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 6262 . . . . . . 7  |-  ( x ( Hom  `  T
) y )  e. 
_V
33 resfunexg 6074 . . . . . . 7  |-  ( ( Fun  1st  /\  (
x ( Hom  `  T
) y )  e. 
_V )  ->  ( 1st  |`  ( x ( Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 670 . . . . . 6  |-  ( 1st  |`  ( x ( Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6807 . . . . 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 5608 . . . . 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 233 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f1stres 6760 . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 756 . . . . . . . . 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 758 . . . . . . . . 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 15678 . . . . . . . 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 2402 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
451, 4, 23, 44, 5, 41, 42xpchom 15665 . . . . . . . . 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 5215 . . . . . . . 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 2443 . . . . . . 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 5656 . . . . . 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 233 . . . . 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 5819 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
5150ad2antrl 726 . . . . . . 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 5819 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 1st `  y
) )
5352ad2antll 727 . . . . . . 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 6252 . . . . . 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 5665 . . . . 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 232 . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 15188 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x ( Hom  `  T
) x ) )
60 fvres 5819 . . . . . . 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 6775 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5809 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2402 . . . . . . . . 9  |-  ( Id
`  D )  =  ( Id `  D
)
68 xp1st 6768 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6769 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 25, 67, 24, 69, 71xpcid 15674 . . . . . . . 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 2443 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5815 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5815 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op1std 6748 . . . . . . 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 2443 . . . . 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 15678 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  P ) x )  =  ( 1st  |`  ( x
( Hom  `  T ) x ) ) )
8079fveq1d 5807 . . . . 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 464 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
8281fveq2d 5809 . . . . 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 2453 . . . 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 1018 . . . . . . . 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 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 ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
86 simp22 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 ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
87 simp23 1032 . . . . . . . 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 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 ) ) )  ->  f  e.  ( x ( Hom  `  T ) y ) )
89 simp3r 1026 . . . . . . . 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 15191 . . . . . . 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 5819 . . . . . . 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 15669 . . . . . 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 2443 . . . . 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 1018 . . . . . . 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 1018 . . . . . . 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 15678 . . . . . 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 5807 . . . . 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 4166 . . . . . . 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 5819 . . . . . . . 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 6252 . . . . . 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 15678 . . . . . . . 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 5807 . . . . . . 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 5819 . . . . . . . 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 2443 . . . . . 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 15678 . . . . . . . 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 5807 . . . . . . 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 5819 . . . . . . . 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 2443 . . . . . 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 6255 . . . . 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 2453 . . . 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 15370 . . 3  |-  ( ph  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  C )
( 2nd `  P
) )
118 df-br 4395 . . 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 196 . 2  |-  ( ph  -> 
<. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  e.  ( T 
Func  C ) )
12022, 119eqeltrd 2490 1  |-  ( ph  ->  P  e.  ( T 
Func  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058   <.cop 3977   class class class wbr 4394    X. cxp 4940    |` cres 4944   Fun wfun 5519    Fn wfn 5520   -->wf 5521   -onto->wfo 5523   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   1stc1st 6736   2ndc2nd 6737   Basecbs 14733   Hom chom 14812  compcco 14813   Catccat 15170   Idccid 15171    Func cfunc 15359    X.c cxpc 15653    1stF c1stf 15654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-fz 11644  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-hom 14825  df-cco 14826  df-cat 15174  df-cid 15175  df-func 15363  df-xpc 15657  df-1stf 15658
This theorem is referenced by:  prf1st  15689  1st2ndprf  15691  uncfcl  15720  uncf1  15721  uncf2  15722  diagcl  15726  diag11  15728  diag12  15729  diag2  15730  yonedalem1  15757  yonedalem21  15758  yonedalem22  15763
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