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Theorem 1stfcl 14277
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 2430 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2430 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 14258 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2430 . . . 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 14271 . . 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 6352 . . . . . . . 8  |-  1st : _V -onto-> _V
11 fofun 5640 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
1210, 11ax-mp 8 . . . . . . 7  |-  Fun  1st
13 fvex 5728 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5728 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 4976 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 5943 . . . . . . 7  |-  ( ( Fun  1st  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 654 . . . . . 6  |-  ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6411 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
(  Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6344 . . . . 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 16 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
(  Hom  `  T ) y ) ) ) )
2120opeq2d 3978 . . 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 2465 . 2  |-  ( ph  ->  P  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  P ) >. )
23 eqid 2430 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
24 eqid 2430 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2430 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
26 eqid 2430 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2430 . . . 4  |-  (comp `  C )  =  (comp `  C )
281, 6, 7xpccat 14270 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f1stres 6354 . . . . 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 2430 . . . . . 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 6092 . . . . . . 7  |-  ( x (  Hom  `  T
) y )  e. 
_V
33 resfunexg 5943 . . . . . . 7  |-  ( ( Fun  1st  /\  (
x (  Hom  `  T
) y )  e. 
_V )  ->  ( 1st  |`  ( x (  Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 654 . . . . . 6  |-  ( 1st  |`  ( x (  Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6406 . . . . 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 5522 . . . . 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 225 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f1stres 6354 . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
42 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
431, 4, 5, 39, 40, 8, 41, 421stf2 14273 . . . . . . . 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 2430 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
451, 4, 23, 44, 5, 41, 42xpchom 14260 . . . . . . . . 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 5132 . . . . . . . 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 2462 . . . . . . 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 5566 . . . . . 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 225 . . . . 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 5731 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
5150ad2antrl 709 . . . . . . 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 5731 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 1st `  y
) )
5352ad2antll 710 . . . . . . 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 6085 . . . . . 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 5574 . . . . 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 224 . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 13890 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x (  Hom  `  T
) x ) )
60 fvres 5731 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  e.  ( x (  Hom  `  T ) x )  ->  ( ( 1st  |`  ( x (  Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 1st `  (
( Id `  T
) `  x )
) )
6159, 60syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( x (  Hom  `  T ) x ) ) `  ( ( Id `  T ) `
 x ) )  =  ( 1st `  (
( Id `  T
) `  x )
) )
62 1st2nd2 6372 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5718 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2430 . . . . . . . . 9  |-  ( Id
`  D )  =  ( Id `  D
)
68 xp1st 6362 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6363 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 25, 67, 24, 69, 71xpcid 14269 . . . . . . . 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 2462 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5728 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5728 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op1std 6343 . . . . . . 7  |-  ( ( ( Id `  T
) `  x )  =  <. ( ( Id
`  C ) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `
 ( 2nd `  x
) ) >.  ->  ( 1st `  ( ( Id
`  T ) `  x ) )  =  ( ( Id `  C ) `  ( 1st `  x ) ) )
7773, 76syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  (
( Id `  T
) `  x )
)  =  ( ( Id `  C ) `
 ( 1st `  x
) ) )
7861, 77eqtrd 2462 . . . . 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 14273 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  P ) x )  =  ( 1st  |`  ( x
(  Hom  `  T ) x ) ) )
8079fveq1d 5716 . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
8281fveq2d 5718 . . . . 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 2472 . . . 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 978 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  T  e.  Cat )
85 simp21 990 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
86 simp22 991 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
87 simp23 992 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
88 simp3l 985 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  f  e.  ( x (  Hom  `  T ) y ) )
89 simp3r 986 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  g  e.  ( y (  Hom  `  T ) z ) )
904, 5, 26, 84, 85, 86, 87, 88, 89catcocl 13893 . . . . . . 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 5731 . . . . . . 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 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 ) ) )  ->  ( ( 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 14264 . . . . . 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 2462 . . . . 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 978 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  C  e.  Cat )
9673ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) )  /\  z  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  /\  ( f  e.  ( x (  Hom  `  T ) y )  /\  g  e.  ( y (  Hom  `  T
) z ) ) )  ->  D  e.  Cat )
971, 4, 5, 95, 96, 8, 85, 871stf2 14273 . . . . . 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 5716 . . . . 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 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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  x )  =  ( 1st `  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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  y )  =  ( 1st `  y
) )
10199, 100opeq12d 3979 . . . . . . 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 5731 . . . . . . . 8  |-  ( z  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 z )  =  ( 1st `  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 ) ) )  ->  ( ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) `  z )  =  ( 1st `  z
) )
104101, 103oveq12d 6085 . . . . . 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 14273 . . . . . . . 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 5716 . . . . . . 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 5731 . . . . . . . 8  |-  ( g  e.  ( y (  Hom  `  T )
z )  ->  (
( 1st  |`  ( y (  Hom  `  T
) z ) ) `
 g )  =  ( 1st `  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 ) ) )  ->  ( ( 1st  |`  ( y (  Hom  `  T )
z ) ) `  g )  =  ( 1st `  g ) )
109106, 108eqtrd 2462 . . . . . 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 14273 . . . . . . . 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 5716 . . . . . . 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 5731 . . . . . . . 8  |-  ( f  e.  ( x (  Hom  `  T )
y )  ->  (
( 1st  |`  ( x (  Hom  `  T
) y ) ) `
 f )  =  ( 1st `  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 ) ) )  ->  ( ( 1st  |`  ( x (  Hom  `  T )
y ) ) `  f )  =  ( 1st `  f ) )
114111, 113eqtrd 2462 . . . . . 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 6088 . . . . 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 2472 . . . 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 14045 . . 3  |-  ( ph  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  C )
( 2nd `  P
) )
118 df-br 4200 . . 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 189 . 2  |-  ( ph  -> 
<. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  e.  ( T 
Func  C ) )
12022, 119eqeltrd 2504 1  |-  ( ph  ->  P  e.  ( T 
Func  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2943   <.cop 3804   class class class wbr 4199    X. cxp 4862    |` cres 4866   Fun wfun 5434    Fn wfn 5435   -->wf 5436   -onto->wfo 5438   ` cfv 5440  (class class class)co 6067    e. cmpt2 6069   1stc1st 6333   2ndc2nd 6334   Basecbs 13452    Hom chom 13523  compcco 13524   Catccat 13872   Idccid 13873    Func cfunc 14034    X.c cxpc 14248    1stF c1stf 14249
This theorem is referenced by:  prf1st  14284  1st2ndprf  14286  uncfcl  14315  uncf1  14316  uncf2  14317  diagcl  14321  diag11  14323  diag12  14324  diag2  14325  yonedalem1  14352  yonedalem21  14353  yonedalem22  14358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-map 7006  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-fz 11028  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-hom 13536  df-cco 13537  df-cat 13876  df-cid 13877  df-func 14038  df-xpc 14252  df-1stf 14253
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