MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stfcl Structured version   Unicode version

Theorem 1stfcl 14332
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 2443 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2443 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
41, 2, 3xpcbas 14313 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
5 eqid 2443 . . . 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 14326 . . 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 6402 . . . . . . . 8  |-  1st : _V -onto-> _V
11 fofun 5689 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
1210, 11ax-mp 5 . . . . . . 7  |-  Fun  1st
13 fvex 5773 . . . . . . . 8  |-  ( Base `  C )  e.  _V
14 fvex 5773 . . . . . . . 8  |-  ( Base `  D )  e.  _V
1513, 14xpex 5025 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  e.  _V
16 resfunexg 5993 . . . . . . 7  |-  ( ( Fun  1st  /\  (
( Base `  C )  X.  ( Base `  D
) )  e.  _V )  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) )  e. 
_V )
1712, 15, 16mp2an 655 . . . . . 6  |-  ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) )  e.  _V
1815, 15mpt2ex 6461 . . . . . 6  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  |->  ( 1st  |`  ( x
(  Hom  `  T ) y ) ) )  e.  _V
1917, 18op2ndd 6394 . . . . 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 4020 . . 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 2478 . 2  |-  ( ph  ->  P  =  <. ( 1st  |`  ( ( Base `  C )  X.  ( Base `  D ) ) ) ,  ( 2nd `  P ) >. )
23 eqid 2443 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
24 eqid 2443 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
25 eqid 2443 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
26 eqid 2443 . . . 4  |-  (comp `  T )  =  (comp `  T )
27 eqid 2443 . . . 4  |-  (comp `  C )  =  (comp `  C )
281, 6, 7xpccat 14325 . . . 4  |-  ( ph  ->  T  e.  Cat )
29 f1stres 6404 . . . . 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 2443 . . . . . 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 6142 . . . . . . 7  |-  ( x (  Hom  `  T
) y )  e. 
_V
33 resfunexg 5993 . . . . . . 7  |-  ( ( Fun  1st  /\  (
x (  Hom  `  T
) y )  e. 
_V )  ->  ( 1st  |`  ( x (  Hom  `  T )
y ) )  e. 
_V )
3412, 32, 33mp2an 655 . . . . . 6  |-  ( 1st  |`  ( x (  Hom  `  T ) y ) )  e.  _V
3531, 34fnmpt2i 6456 . . . . 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 5571 . . . . 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 226 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( ( ( Base `  C
)  X.  ( Base `  D ) )  X.  ( ( Base `  C
)  X.  ( Base `  D ) ) ) )
38 f1stres 6404 . . . . . 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 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  C  e.  Cat )
407adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  D ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  D
) ) ) )  ->  D  e.  Cat )
41 simprl 734 . . . . . . . . 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 735 . . . . . . . . 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 14328 . . . . . . . 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 2443 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
451, 4, 23, 44, 5, 41, 42xpchom 14315 . . . . . . . . 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 5181 . . . . . . . 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 2475 . . . . . . 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 5615 . . . . . 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 226 . . . . 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 5776 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
5150ad2antrl 710 . . . . . . 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 5776 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 y )  =  ( 1st `  y
) )
5352ad2antll 711 . . . . . . 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 6135 . . . . . 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 5623 . . . . 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 225 . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  T  e.  Cat )
58 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
594, 5, 24, 57, 58catidcl 13945 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  e.  ( x (  Hom  `  T
) x ) )
60 fvres 5776 . . . . . . 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 6422 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6362adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
6463fveq2d 5767 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  ( ( Id `  T
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
656adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  C  e.  Cat )
667adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  D  e.  Cat )
67 eqid 2443 . . . . . . . . 9  |-  ( Id
`  D )  =  ( Id `  D
)
68 xp1st 6412 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
6968adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
70 xp2nd 6413 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  D ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
7170adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  D ) )
721, 65, 66, 2, 3, 25, 67, 24, 69, 71xpcid 14324 . . . . . . . 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 2475 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( Id
`  T ) `  x )  =  <. ( ( Id `  C
) `  ( 1st `  x ) ) ,  ( ( Id `  D ) `  ( 2nd `  x ) )
>. )
74 fvex 5773 . . . . . . . 8  |-  ( ( Id `  C ) `
 ( 1st `  x
) )  e.  _V
75 fvex 5773 . . . . . . . 8  |-  ( ( Id `  D ) `
 ( 2nd `  x
) )  e.  _V
7674, 75op1std 6393 . . . . . . 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 2475 . . . . 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 14328 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( x ( 2nd `  P ) x )  =  ( 1st  |`  ( x
(  Hom  `  T ) x ) ) )
8079fveq1d 5765 . . . . 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 454 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )  ->  ( ( 1st  |`  ( ( Base `  C
)  X.  ( Base `  D ) ) ) `
 x )  =  ( 1st `  x
) )
8281fveq2d 5767 . . . . 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 2485 . . . 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 979 . . . . . . . 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 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 ) ) )  ->  x  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
86 simp22 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 ) ) )  ->  y  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
87 simp23 993 . . . . . . . 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 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 ) ) )  ->  f  e.  ( x (  Hom  `  T ) y ) )
89 simp3r 987 . . . . . . . 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 13948 . . . . . . 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 5776 . . . . . . 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 14319 . . . . . 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 2475 . . . . 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 979 . . . . . . 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 979 . . . . . . 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 14328 . . . . . 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 5765 . . . . 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 4021 . . . . . . 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 5776 . . . . . . . 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 6135 . . . . . 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 14328 . . . . . . . 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 5765 . . . . . . 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 5776 . . . . . . . 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 2475 . . . . . 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 14328 . . . . . . . 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 5765 . . . . . . 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 5776 . . . . . . . 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 2475 . . . . . 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 6138 . . . . 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 2485 . . . 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 14100 . . 3  |-  ( ph  ->  ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ( T  Func  C )
( 2nd `  P
) )
118 df-br 4244 . . 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 190 . 2  |-  ( ph  -> 
<. ( 1st  |`  (
( Base `  C )  X.  ( Base `  D
) ) ) ,  ( 2nd `  P
) >.  e.  ( T 
Func  C ) )
12022, 119eqeltrd 2517 1  |-  ( ph  ->  P  e.  ( T 
Func  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   _Vcvv 2965   <.cop 3846   class class class wbr 4243    X. cxp 4911    |` cres 4915   Fun wfun 5483    Fn wfn 5484   -->wf 5485   -onto->wfo 5487   ` cfv 5489  (class class class)co 6117    e. cmpt2 6119   1stc1st 6383   2ndc2nd 6384   Basecbs 13507    Hom chom 13578  compcco 13579   Catccat 13927   Idccid 13928    Func cfunc 14089    X.c cxpc 14303    1stF c1stf 14304
This theorem is referenced by:  prf1st  14339  1st2ndprf  14341  uncfcl  14370  uncf1  14371  uncf2  14372  diagcl  14376  diag11  14378  diag12  14379  diag2  14380  yonedalem1  14407  yonedalem21  14408  yonedalem22  14413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-map 7056  df-ixp 7100  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-4 10098  df-5 10099  df-6 10100  df-7 10101  df-8 10102  df-9 10103  df-10 10104  df-n0 10260  df-z 10321  df-dec 10421  df-uz 10527  df-fz 11082  df-struct 13509  df-ndx 13510  df-slot 13511  df-base 13512  df-hom 13591  df-cco 13592  df-cat 13931  df-cid 13932  df-func 14093  df-xpc 14307  df-1stf 14308
  Copyright terms: Public domain W3C validator