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Theorem 1st2ndprf 15028
Description: Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
1st2ndprf.t  |-  T  =  ( D  X.c  E )
1st2ndprf.f  |-  ( ph  ->  F  e.  ( C 
Func  T ) )
1st2ndprf.d  |-  ( ph  ->  D  e.  Cat )
1st2ndprf.e  |-  ( ph  ->  E  e.  Cat )
Assertion
Ref Expression
1st2ndprf  |-  ( ph  ->  F  =  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func 
F ) ) )

Proof of Theorem 1st2ndprf
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 1st2ndprf.t . . . . . . 7  |-  T  =  ( D  X.c  E )
3 eqid 2443 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
4 eqid 2443 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
52, 3, 4xpcbas 15000 . . . . . 6  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  T )
6 relfunc 14784 . . . . . . 7  |-  Rel  ( C  Func  T )
7 1st2ndprf.f . . . . . . 7  |-  ( ph  ->  F  e.  ( C 
Func  T ) )
8 1st2ndbr 6635 . . . . . . 7  |-  ( ( Rel  ( C  Func  T )  /\  F  e.  ( C  Func  T
) )  ->  ( 1st `  F ) ( C  Func  T )
( 2nd `  F
) )
96, 7, 8sylancr 663 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  T ) ( 2nd `  F
) )
101, 5, 9funcf1 14788 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
1110feqmptd 5756 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
1210ffvelrnda 5855 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
13 1st2nd2 6625 . . . . . . 7  |-  ( ( ( 1st `  F
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  -> 
( ( 1st `  F
) `  x )  =  <. ( 1st `  (
( 1st `  F
) `  x )
) ,  ( 2nd `  ( ( 1st `  F
) `  x )
) >. )
1412, 13syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  =  <. ( 1st `  ( ( 1st `  F ) `
 x ) ) ,  ( 2nd `  (
( 1st `  F
) `  x )
) >. )
157adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F  e.  ( C  Func  T ) )
16 1st2ndprf.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  Cat )
17 1st2ndprf.e . . . . . . . . . . 11  |-  ( ph  ->  E  e.  Cat )
18 eqid 2443 . . . . . . . . . . 11  |-  ( D  1stF  E )  =  ( D  1stF  E )
192, 16, 17, 181stfcl 15019 . . . . . . . . . 10  |-  ( ph  ->  ( D  1stF  E )  e.  ( T  Func  D
) )
2019adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( D  1stF  E )  e.  ( T 
Func  D ) )
21 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
221, 15, 20, 21cofu1 14806 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  1stF  E )  o.func  F )
) `  x )  =  ( ( 1st `  ( D  1stF  E )
) `  ( ( 1st `  F ) `  x ) ) )
23 eqid 2443 . . . . . . . . 9  |-  ( Hom  `  T )  =  ( Hom  `  T )
2416adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2517adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
262, 5, 23, 24, 25, 18, 121stf1 15014 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  ( ( 1st `  F ) `
 x ) )  =  ( 1st `  (
( 1st `  F
) `  x )
) )
2722, 26eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  1stF  E )  o.func  F )
) `  x )  =  ( 1st `  (
( 1st `  F
) `  x )
) )
28 eqid 2443 . . . . . . . . . . 11  |-  ( D  2ndF  E )  =  ( D  2ndF  E )
292, 16, 17, 282ndfcl 15020 . . . . . . . . . 10  |-  ( ph  ->  ( D  2ndF  E )  e.  ( T  Func  E
) )
3029adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( D  2ndF  E )  e.  ( T 
Func  E ) )
311, 15, 30, 21cofu1 14806 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x )  =  ( ( 1st `  ( D  2ndF  E )
) `  ( ( 1st `  F ) `  x ) ) )
322, 5, 23, 24, 25, 28, 122ndf1 15017 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  2ndF  E ) ) `  ( ( 1st `  F ) `
 x ) )  =  ( 2nd `  (
( 1st `  F
) `  x )
) )
3331, 32eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x )  =  ( 2nd `  (
( 1st `  F
) `  x )
) )
3427, 33opeq12d 4079 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  ( ( D  1stF  E )  o.func  F )
) `  x ) ,  ( ( 1st `  ( ( D  2ndF  E )  o.func 
F ) ) `  x ) >.  =  <. ( 1st `  ( ( 1st `  F ) `
 x ) ) ,  ( 2nd `  (
( 1st `  F
) `  x )
) >. )
3514, 34eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  =  <. ( ( 1st `  (
( D  1stF  E )  o.func  F ) ) `  x
) ,  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x ) >. )
3635mpteq2dva 4390 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( 1st `  F
) `  x )
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  ( ( D  1stF  E )  o.func 
F ) ) `  x ) ,  ( ( 1st `  (
( D  2ndF  E )  o.func  F ) ) `  x
) >. ) )
3711, 36eqtrd 2475 . . 3  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  ( ( D  1stF  E )  o.func 
F ) ) `  x ) ,  ( ( 1st `  (
( D  2ndF  E )  o.func  F ) ) `  x
) >. ) )
381, 9funcfn2 14791 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
39 fnov 6210 . . . . 5  |-  ( ( 2nd `  F )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
4038, 39sylib 196 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
41 eqid 2443 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
429adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  T )
( 2nd `  F
) )
43 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
44 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
451, 41, 23, 42, 43, 44funcf2 14790 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y ) : ( x ( Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) ) )
4645feqmptd 5756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y )  =  ( f  e.  ( x ( Hom  `  C
) y )  |->  ( ( x ( 2nd `  F ) y ) `
 f ) ) )
472, 23relxpchom 15003 . . . . . . . . . 10  |-  Rel  (
( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
)
4845ffvelrnda 5855 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  F ) y ) `  f
)  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) )
49 1st2nd 6632 . . . . . . . . . 10  |-  ( ( Rel  ( ( ( 1st `  F ) `
 x ) ( Hom  `  T )
( ( 1st `  F
) `  y )
)  /\  ( (
x ( 2nd `  F
) y ) `  f )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) )  ->  (
( x ( 2nd `  F ) y ) `
 f )  = 
<. ( 1st `  (
( x ( 2nd `  F ) y ) `
 f ) ) ,  ( 2nd `  (
( x ( 2nd `  F ) y ) `
 f ) )
>. )
5047, 48, 49sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  F ) y ) `  f
)  =  <. ( 1st `  ( ( x ( 2nd `  F
) y ) `  f ) ) ,  ( 2nd `  (
( x ( 2nd `  F ) y ) `
 f ) )
>. )
517ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  F  e.  ( C  Func  T ) )
5219ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( D  1stF  E )  e.  ( T  Func  D
) )
5343adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
5444adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
55 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
f  e.  ( x ( Hom  `  C
) y ) )
561, 51, 52, 53, 54, 41, 55cofu2 14808 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( ( D  1stF  E )  o.func  F )
) y ) `  f )  =  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) ) )
5716adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  D  e.  Cat )
5817adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  E  e.  Cat )
5912adantrr 716 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6010ffvelrnda 5855 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6160adantrl 715 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
622, 5, 23, 57, 58, 18, 59, 611stf2 15015 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  F
) `  x )
( 2nd `  ( D  1stF  E ) ) ( ( 1st `  F
) `  y )
)  =  ( 1st  |`  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) ) ) )
6362adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( ( 1st `  F ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  F
) `  y )
)  =  ( 1st  |`  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) ) ) )
6463fveq1d 5705 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  F ) `
 x ) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) )  =  ( ( 1st  |`  (
( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
65 fvres 5716 . . . . . . . . . . . 12  |-  ( ( ( x ( 2nd `  F ) y ) `
 f )  e.  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) )  ->  (
( 1st  |`  ( ( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) )  =  ( 1st `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
6648, 65syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( 1st  |`  (
( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) )  =  ( 1st `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
6756, 64, 663eqtrd 2479 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( ( D  1stF  E )  o.func  F )
) y ) `  f )  =  ( 1st `  ( ( x ( 2nd `  F
) y ) `  f ) ) )
6829ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( D  2ndF  E )  e.  ( T  Func  E
) )
691, 51, 68, 53, 54, 41, 55cofu2 14808 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f )  =  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) ) )
702, 5, 23, 57, 58, 28, 59, 612ndf2 15018 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  F
) `  x )
( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  F
) `  y )
)  =  ( 2nd  |`  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) ) ) )
7170adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( ( 1st `  F ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  F
) `  y )
)  =  ( 2nd  |`  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) ) ) )
7271fveq1d 5705 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  F ) `
 x ) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) )  =  ( ( 2nd  |`  (
( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
73 fvres 5716 . . . . . . . . . . . 12  |-  ( ( ( x ( 2nd `  F ) y ) `
 f )  e.  ( ( ( 1st `  F ) `  x
) ( Hom  `  T
) ( ( 1st `  F ) `  y
) )  ->  (
( 2nd  |`  ( ( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) )  =  ( 2nd `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
7448, 73syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( 2nd  |`  (
( ( 1st `  F
) `  x )
( Hom  `  T ) ( ( 1st `  F
) `  y )
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) )  =  ( 2nd `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
7569, 72, 743eqtrd 2479 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f )  =  ( 2nd `  ( ( x ( 2nd `  F
) y ) `  f ) ) )
7667, 75opeq12d 4079 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  <. ( ( x ( 2nd `  ( ( D  1stF  E )  o.func  F )
) y ) `  f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func 
F ) ) y ) `  f )
>.  =  <. ( 1st `  ( ( x ( 2nd `  F ) y ) `  f
) ) ,  ( 2nd `  ( ( x ( 2nd `  F
) y ) `  f ) ) >.
)
7750, 76eqtr4d 2478 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  F ) y ) `  f
)  =  <. (
( x ( 2nd `  ( ( D  1stF  E )  o.func 
F ) ) y ) `  f ) ,  ( ( x ( 2nd `  (
( D  2ndF  E )  o.func  F ) ) y ) `
 f ) >.
)
7877mpteq2dva 4390 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
f  e.  ( x ( Hom  `  C
) y )  |->  ( ( x ( 2nd `  F ) y ) `
 f ) )  =  ( f  e.  ( x ( Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  (
( D  1stF  E )  o.func  F ) ) y ) `
 f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f ) >. )
)
7946, 78eqtrd 2475 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y )  =  ( f  e.  ( x ( Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  ( ( D  1stF  E )  o.func  F )
) y ) `  f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func 
F ) ) y ) `  f )
>. ) )
80793impb 1183 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  F ) y )  =  ( f  e.  ( x ( Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  ( ( D  1stF  E )  o.func  F )
) y ) `  f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func 
F ) ) y ) `  f )
>. ) )
8180mpt2eq3dva 6162 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( x ( 2nd `  F
) y ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( f  e.  ( x ( Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  (
( D  1stF  E )  o.func  F ) ) y ) `
 f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f ) >. )
) )
8240, 81eqtrd 2475 . . 3  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( f  e.  ( x ( Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  (
( D  1stF  E )  o.func  F ) ) y ) `
 f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f ) >. )
) )
8337, 82opeq12d 4079 . 2  |-  ( ph  -> 
<. ( 1st `  F
) ,  ( 2nd `  F ) >.  =  <. ( x  e.  ( Base `  C )  |->  <. (
( 1st `  (
( D  1stF  E )  o.func  F ) ) `  x
) ,  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x ) >. ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( f  e.  ( x ( Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  (
( D  1stF  E )  o.func  F ) ) y ) `
 f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f ) >. )
) >. )
84 1st2nd 6632 . . 3  |-  ( ( Rel  ( C  Func  T )  /\  F  e.  ( C  Func  T
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
856, 7, 84sylancr 663 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
86 eqid 2443 . . 3  |-  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func 
F ) )  =  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func  F )
)
877, 19cofucl 14810 . . 3  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
F )  e.  ( C  Func  D )
)
887, 29cofucl 14810 . . 3  |-  ( ph  ->  ( ( D  2ndF  E )  o.func 
F )  e.  ( C  Func  E )
)
8986, 1, 41, 87, 88prfval 15021 . 2  |-  ( ph  ->  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func  F )
)  =  <. (
x  e.  ( Base `  C )  |->  <. (
( 1st `  (
( D  1stF  E )  o.func  F ) ) `  x
) ,  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x ) >. ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( f  e.  ( x ( Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  (
( D  1stF  E )  o.func  F ) ) y ) `
 f ) ,  ( ( x ( 2nd `  ( ( D  2ndF  E )  o.func  F )
) y ) `  f ) >. )
) >. )
9083, 85, 893eqtr4d 2485 1  |-  ( ph  ->  F  =  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func 
F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3895   class class class wbr 4304    e. cmpt 4362    X. cxp 4850    |` cres 4854   Rel wrel 4857    Fn wfn 5425   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   1stc1st 6587   2ndc2nd 6588   Basecbs 14186   Hom chom 14261   Catccat 14614    Func cfunc 14776    o.func ccofu 14778    X.c cxpc 14990    1stF c1stf 14991    2ndF c2ndf 14992   ⟨,⟩F cprf 14993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-hom 14274  df-cco 14275  df-cat 14618  df-cid 14619  df-func 14780  df-cofu 14782  df-xpc 14994  df-1stf 14995  df-2ndf 14996  df-prf 14997
This theorem is referenced by: (None)
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