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Theorem 1st2ndprf 14296
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 2436 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 1st2ndprf.t . . . . . . 7  |-  T  =  ( D  X.c  E )
3 eqid 2436 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
4 eqid 2436 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
52, 3, 4xpcbas 14268 . . . . . 6  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  T )
6 relfunc 14052 . . . . . . 7  |-  Rel  ( C  Func  T )
7 1st2ndprf.f . . . . . . 7  |-  ( ph  ->  F  e.  ( C 
Func  T ) )
8 1st2ndbr 6389 . . . . . . 7  |-  ( ( Rel  ( C  Func  T )  /\  F  e.  ( C  Func  T
) )  ->  ( 1st `  F ) ( C  Func  T )
( 2nd `  F
) )
96, 7, 8sylancr 645 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  T ) ( 2nd `  F
) )
101, 5, 9funcf1 14056 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
1110feqmptd 5772 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
1210ffvelrnda 5863 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
13 1st2nd2 6379 . . . . . . 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 452 . . . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( D  1stF  E )  =  ( D  1stF  E )
192, 16, 17, 181stfcl 14287 . . . . . . . . . 10  |-  ( ph  ->  ( D  1stF  E )  e.  ( T  Func  D
) )
2019adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( D  1stF  E )  e.  ( T 
Func  D ) )
21 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
221, 15, 20, 21cofu1 14074 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  1stF  E )  o.func  F )
) `  x )  =  ( ( 1st `  ( D  1stF  E )
) `  ( ( 1st `  F ) `  x ) ) )
23 eqid 2436 . . . . . . . . 9  |-  (  Hom  `  T )  =  (  Hom  `  T )
2416adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2517adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
262, 5, 23, 24, 25, 18, 121stf1 14282 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  ( ( 1st `  F ) `
 x ) )  =  ( 1st `  (
( 1st `  F
) `  x )
) )
2722, 26eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  1stF  E )  o.func  F )
) `  x )  =  ( 1st `  (
( 1st `  F
) `  x )
) )
28 eqid 2436 . . . . . . . . . . 11  |-  ( D  2ndF  E )  =  ( D  2ndF  E )
292, 16, 17, 282ndfcl 14288 . . . . . . . . . 10  |-  ( ph  ->  ( D  2ndF  E )  e.  ( T  Func  E
) )
3029adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( D  2ndF  E )  e.  ( T 
Func  E ) )
311, 15, 30, 21cofu1 14074 . . . . . . . 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 14285 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  2ndF  E ) ) `  ( ( 1st `  F ) `
 x ) )  =  ( 2nd `  (
( 1st `  F
) `  x )
) )
3331, 32eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( D  2ndF  E )  o.func  F )
) `  x )  =  ( 2nd `  (
( 1st `  F
) `  x )
) )
3427, 33opeq12d 3985 . . . . . 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 2471 . . . . 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 4288 . . . 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 2468 . . 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 14059 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
39 fnov 6171 . . . . 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 189 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
41 eqid 2436 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
429adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  T )
( 2nd `  F
) )
43 simprl 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
44 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
451, 41, 23, 42, 43, 44funcf2 14058 . . . . . . . 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 5772 . . . . . . 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 14271 . . . . . . . . . 10  |-  Rel  (
( ( 1st `  F
) `  x )
(  Hom  `  T ) ( ( 1st `  F
) `  y )
)
4845ffvelrnda 5863 . . . . . . . . . 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 6386 . . . . . . . . . 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 645 . . . . . . . . 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 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  F  e.  ( C  Func  T ) )
5219ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( D  1stF  E )  e.  ( T  Func  D
) )
5343adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
5444adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
55 simpr 448 . . . . . . . . . . . 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 14076 . . . . . . . . . . 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 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  D  e.  Cat )
5817adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  E  e.  Cat )
5912adantrr 698 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6010ffvelrnda 5863 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6160adantrl 697 . . . . . . . . . . . . . 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 14283 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 5723 . . . . . . . . . . 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 5738 . . . . . . . . . . . 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 2472 . . . . . . . . . 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 707 . . . . . . . . . . . 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 14076 . . . . . . . . . . 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 14286 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 5723 . . . . . . . . . . 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 5738 . . . . . . . . . . . 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 2472 . . . . . . . . . 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 3985 . . . . . . . . 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 2471 . . . . . . . 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 4288 . . . . . . 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 2468 . . . . . 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 1149 . . . . 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 6131 . . . 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 2468 . . 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 3985 . 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 6386 . . 3  |-  ( ( Rel  ( C  Func  T )  /\  F  e.  ( C  Func  T
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
856, 7, 84sylancr 645 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
86 eqid 2436 . . 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 14078 . . 3  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
F )  e.  ( C  Func  D )
)
887, 29cofucl 14078 . . 3  |-  ( ph  ->  ( ( D  2ndF  E )  o.func 
F )  e.  ( C  Func  E )
)
8986, 1, 41, 87, 88prfval 14289 . 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 2478 1  |-  ( ph  ->  F  =  ( ( ( D  1stF  E )  o.func  F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func 
F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3810   class class class wbr 4205    e. cmpt 4259    X. cxp 4869    |` cres 4873   Rel wrel 4876    Fn wfn 5442   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   1stc1st 6340   2ndc2nd 6341   Basecbs 13462    Hom chom 13533   Catccat 13882    Func cfunc 14044    o.func ccofu 14046    X.c cxpc 14258    1stF c1stf 14259    2ndF c2ndf 14260   ⟨,⟩F cprf 14261
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-map 7013  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-fz 11037  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-hom 13546  df-cco 13547  df-cat 13886  df-cid 13887  df-func 14048  df-cofu 14050  df-xpc 14262  df-1stf 14263  df-2ndf 14264  df-prf 14265
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