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Theorem prfcl 14302
Description: The pairing of functors  F : C
--> D and  G : C --> D is a functor  <. F ,  G >. : C --> ( D  X.  E ). (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfcl.p  |-  P  =  ( F ⟨,⟩F  G )
prfcl.t  |-  T  =  ( D  X.c  E )
prfcl.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prfcl.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
Assertion
Ref Expression
prfcl  |-  ( ph  ->  P  e.  ( C 
Func  T ) )

Proof of Theorem prfcl
Dummy variables  f 
g  h  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfcl.p . . . 4  |-  P  =  ( F ⟨,⟩F  G )
2 eqid 2438 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2438 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 prfcl.c . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 prfcl.d . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
61, 2, 3, 4, 5prfval 14298 . . 3  |-  ( ph  ->  P  =  <. (
x  e.  ( Base `  C )  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >. )
7 fvex 5744 . . . . . . 7  |-  ( Base `  C )  e.  _V
87mptex 5968 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
97, 7mpt2ex 6427 . . . . . 6  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )  e.  _V
108, 9op1std 6359 . . . . 5  |-  ( P  =  <. ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >.  ->  ( 1st `  P )  =  ( x  e.  (
Base `  C )  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
116, 10syl 16 . . . 4  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
128, 9op2ndd 6360 . . . . 5  |-  ( P  =  <. ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >.  ->  ( 2nd `  P )  =  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( h  e.  ( x (  Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) )
136, 12syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) )
1411, 13opeq12d 3994 . . 3  |-  ( ph  -> 
<. ( 1st `  P
) ,  ( 2nd `  P ) >.  =  <. ( x  e.  ( Base `  C )  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >. )
156, 14eqtr4d 2473 . 2  |-  ( ph  ->  P  =  <. ( 1st `  P ) ,  ( 2nd `  P
) >. )
16 prfcl.t . . . . 5  |-  T  =  ( D  X.c  E )
17 eqid 2438 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
18 eqid 2438 . . . . 5  |-  ( Base `  E )  =  (
Base `  E )
1916, 17, 18xpcbas 14277 . . . 4  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  T )
20 eqid 2438 . . . 4  |-  (  Hom  `  T )  =  (  Hom  `  T )
21 eqid 2438 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
22 eqid 2438 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
23 eqid 2438 . . . 4  |-  (comp `  C )  =  (comp `  C )
24 eqid 2438 . . . 4  |-  (comp `  T )  =  (comp `  T )
25 funcrcl 14062 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
264, 25syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2726simpld 447 . . . 4  |-  ( ph  ->  C  e.  Cat )
2826simprd 451 . . . . 5  |-  ( ph  ->  D  e.  Cat )
29 funcrcl 14062 . . . . . . 7  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
305, 29syl 16 . . . . . 6  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
3130simprd 451 . . . . 5  |-  ( ph  ->  E  e.  Cat )
3216, 28, 31xpccat 14289 . . . 4  |-  ( ph  ->  T  e.  Cat )
33 relfunc 14061 . . . . . . . . . 10  |-  Rel  ( C  Func  D )
34 1st2ndbr 6398 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3533, 4, 34sylancr 646 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
362, 17, 35funcf1 14065 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
3736ffvelrnda 5872 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
38 relfunc 14061 . . . . . . . . . 10  |-  Rel  ( C  Func  E )
39 1st2ndbr 6398 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
4038, 5, 39sylancr 646 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
412, 18, 40funcf1 14065 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
4241ffvelrnda 5872 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
43 opelxpi 4912 . . . . . . 7  |-  ( ( ( ( 1st `  F
) `  x )  e.  ( Base `  D
)  /\  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
4437, 42, 43syl2anc 644 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
45 eqid 2438 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4644, 45fmptd 5895 . . . . 5  |-  ( ph  ->  ( x  e.  (
Base `  C )  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) : (
Base `  C ) --> ( ( Base `  D
)  X.  ( Base `  E ) ) )
4711feq1d 5582 . . . . 5  |-  ( ph  ->  ( ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) )  <->  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) ) )
4846, 47mpbird 225 . . . 4  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
49 eqid 2438 . . . . . 6  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )
50 ovex 6108 . . . . . . 7  |-  ( x (  Hom  `  C
) y )  e. 
_V
5150mptex 5968 . . . . . 6  |-  ( h  e.  ( x (  Hom  `  C )
y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. )  e.  _V
5249, 51fnmpt2i 6422 . . . . 5  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
5313fneq1d 5538 . . . . 5  |-  ( ph  ->  ( ( 2nd `  P
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) )  <->  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )  Fn  (
( Base `  C )  X.  ( Base `  C
) ) ) )
5452, 53mpbiri 226 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
55 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
5635adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
57 simprl 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
58 simprr 735 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
592, 3, 55, 56, 57, 58funcf2 14067 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
6059ffvelrnda 5872 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  h  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  F ) y ) `  h
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
61 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  E )  =  (  Hom  `  E )
6240adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
632, 3, 61, 62, 57, 58funcf2 14067 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  G
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  G ) `  x
) (  Hom  `  E
) ( ( 1st `  G ) `  y
) ) )
6463ffvelrnda 5872 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  h  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  G ) y ) `  h
)  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) )
65 opelxpi 4912 . . . . . . . 8  |-  ( ( ( ( x ( 2nd `  F ) y ) `  h
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  /\  ( (
x ( 2nd `  G
) y ) `  h )  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) )  ->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >.  e.  ( ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) ) )
6660, 64, 65syl2anc 644 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  h  e.  ( x
(  Hom  `  C ) y ) )  ->  <. ( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >.  e.  ( ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) )  X.  (
( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) ) )
674adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C  Func  D
) )
685adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( C  Func  E
) )
691, 2, 3, 67, 68, 57prf1 14299 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
701, 2, 3, 67, 68, 58prf1 14299 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  =  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >.
)
7169, 70oveq12d 6101 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
)  =  ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (  Hom  `  T
) <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >.
) )
7237adantrr 699 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
7342adantrr 699 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  E
) )
7436ffvelrnda 5872 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
7574adantrl 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
7641ffvelrnda 5872 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  E )
)
7776adantrl 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  E
) )
7816, 17, 18, 55, 61, 72, 73, 75, 77, 20xpchom2 14285 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (  Hom  `  T
) <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >.
)  =  ( ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) ) )
7971, 78eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
)  =  ( ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) ) )
8079adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  h  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( 1st `  P ) `  x
) (  Hom  `  T
) ( ( 1st `  P ) `  y
) )  =  ( ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) )  X.  (
( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) ) )
8166, 80eleqtrrd 2515 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  h  e.  ( x
(  Hom  `  C ) y ) )  ->  <. ( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >.  e.  ( ( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
) )
82 eqid 2438 . . . . . 6  |-  ( h  e.  ( x (  Hom  `  C )
y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. )  =  ( h  e.  ( x (  Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
8381, 82fmptd 5895 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
h  e.  ( x (  Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
) )
8413oveqd 6100 . . . . . . 7  |-  ( ph  ->  ( x ( 2nd `  P ) y )  =  ( x ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) y ) )
8549ovmpt4g 6198 . . . . . . . 8  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
)  e.  _V )  ->  ( x ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) y )  =  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )
8651, 85mp3an3 1269 . . . . . . 7  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) )  ->  (
x ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) y )  =  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )
8784, 86sylan9eq 2490 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  P
) y )  =  ( h  e.  ( x (  Hom  `  C
) y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
)
8887feq1d 5582 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( x ( 2nd `  P ) y ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
)  <->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) : ( x (  Hom  `  C
) y ) --> ( ( ( 1st `  P
) `  x )
(  Hom  `  T ) ( ( 1st `  P
) `  y )
) ) )
8983, 88mpbird 225 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  P
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  P ) `  x
) (  Hom  `  T
) ( ( 1st `  P ) `  y
) ) )
90 eqid 2438 . . . . . . 7  |-  ( Id
`  D )  =  ( Id `  D
)
9135adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
92 simpr 449 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
932, 21, 90, 91, 92funcid 14069 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  D
) `  ( ( 1st `  F ) `  x ) ) )
94 eqid 2438 . . . . . . 7  |-  ( Id
`  E )  =  ( Id `  E
)
9540adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  G ) ( C 
Func  E ) ( 2nd `  G ) )
962, 21, 94, 95, 92funcid 14069 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  G
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  E
) `  ( ( 1st `  G ) `  x ) ) )
9793, 96opeq12d 3994 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) ) ,  ( ( x ( 2nd `  G ) x ) `
 ( ( Id
`  C ) `  x ) ) >.  =  <. ( ( Id
`  D ) `  ( ( 1st `  F
) `  x )
) ,  ( ( Id `  E ) `
 ( ( 1st `  G ) `  x
) ) >. )
984adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F  e.  ( C  Func  D ) )
995adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  G  e.  ( C  Func  E ) )
10027adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
1012, 3, 21, 100, 92catidcl 13909 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
1021, 2, 3, 98, 99, 92, 92, 101prf2 14301 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  P
) x ) `  ( ( Id `  C ) `  x
) )  =  <. ( ( x ( 2nd `  F ) x ) `
 ( ( Id
`  C ) `  x ) ) ,  ( ( x ( 2nd `  G ) x ) `  (
( Id `  C
) `  x )
) >. )
1031, 2, 3, 98, 99, 92prf1 14299 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  =  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
104103fveq2d 5734 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  ( ( Id `  T ) `
 <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
10528adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
10631adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
10716, 105, 106, 17, 18, 90, 94, 22, 37, 42xpcid 14288 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  <. ( ( Id `  D
) `  ( ( 1st `  F ) `  x ) ) ,  ( ( Id `  E ) `  (
( 1st `  G
) `  x )
) >. )
108104, 107eqtrd 2470 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  <. (
( Id `  D
) `  ( ( 1st `  F ) `  x ) ) ,  ( ( Id `  E ) `  (
( 1st `  G
) `  x )
) >. )
10997, 102, 1083eqtr4d 2480 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  P
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  T
) `  ( ( 1st `  P ) `  x ) ) )
110 eqid 2438 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
111353ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
112 simp21 991 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  x  e.  ( Base `  C )
)
113 simp22 992 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  y  e.  ( Base `  C )
)
114 simp23 993 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  z  e.  ( Base `  C )
)
115 simp3l 986 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
116 simp3r 987 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  g  e.  ( y (  Hom  `  C ) z ) )
1172, 3, 23, 110, 111, 112, 113, 114, 115, 116funcco 14070 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  F
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  F ) z ) `
 g ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
118 eqid 2438 . . . . . . 7  |-  (comp `  E )  =  (comp `  E )
11953ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  G  e.  ( C  Func  E ) )
12038, 119, 39sylancr 646 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( 1st `  G ) ( C 
Func  E ) ( 2nd `  G ) )
1212, 3, 23, 118, 120, 112, 113, 114, 115, 116funcco 14070 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  G
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  G ) z ) `
 g ) (
<. ( ( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  E
) ( ( 1st `  G ) `  z
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) )
122117, 121opeq12d 3994 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  <. ( ( x ( 2nd `  F
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) ) ,  ( ( x ( 2nd `  G
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) ) >.  =  <. ( ( ( y ( 2nd `  F ) z ) `  g
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) ,  ( ( ( y ( 2nd `  G
) z ) `  g ) ( <.
( ( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  E
) ( ( 1st `  G ) `  z
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) >.
)
12343ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  F  e.  ( C  Func  D ) )
124273ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  C  e.  Cat )
1252, 3, 23, 124, 112, 113, 114, 115, 116catcocl 13912 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (  Hom  `  C )
z ) )
1261, 2, 3, 123, 119, 112, 114, 125prf2 14301 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  P
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  <. (
( x ( 2nd `  F ) z ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) ) ,  ( ( x ( 2nd `  G
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) ) >. )
1271, 2, 3, 123, 119, 112prf1 14299 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  P ) `  x )  =  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
1281, 2, 3, 123, 119, 113prf1 14299 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  P ) `  y )  =  <. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. )
129127, 128opeq12d 3994 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  <. ( ( 1st `  P ) `
 x ) ,  ( ( 1st `  P
) `  y ) >.  =  <. <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >. ,  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >. >. )
1301, 2, 3, 123, 119, 114prf1 14299 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  P ) `  z )  =  <. ( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. )
131129, 130oveq12d 6101 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( <. ( ( 1st `  P
) `  x ) ,  ( ( 1st `  P ) `  y
) >. (comp `  T
) ( ( 1st `  P ) `  z
) )  =  (
<. <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >. ,  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >. >. (comp `  T ) <. ( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. ) )
1321, 2, 3, 123, 119, 113, 114, 116prf2 14301 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
y ( 2nd `  P
) z ) `  g )  =  <. ( ( y ( 2nd `  F ) z ) `
 g ) ,  ( ( y ( 2nd `  G ) z ) `  g
) >. )
1331, 2, 3, 123, 119, 112, 113, 115prf2 14301 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  P
) y ) `  f )  =  <. ( ( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. )
134131, 132, 133oveq123d 6104 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
( y ( 2nd `  P ) z ) `
 g ) (
<. ( ( 1st `  P
) `  x ) ,  ( ( 1st `  P ) `  y
) >. (comp `  T
) ( ( 1st `  P ) `  z
) ) ( ( x ( 2nd `  P
) y ) `  f ) )  =  ( <. ( ( y ( 2nd `  F
) z ) `  g ) ,  ( ( y ( 2nd `  G ) z ) `
 g ) >.
( <. <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >. ,  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >. >. (comp `  T ) <. ( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. ) <. (
( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. ) )
135363ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
136135, 112ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
137413ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( 1st `  G ) : (
Base `  C ) --> ( Base `  E )
)
138137, 112ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
139135, 113ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
140137, 113ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  E )
)
141135, 114ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  F ) `  z )  e.  (
Base `  D )
)
142137, 114ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( ( 1st `  G ) `  z )  e.  (
Base `  E )
)
1432, 3, 55, 111, 112, 113funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( x
( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
144143, 115ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  F
) y ) `  f )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
1452, 3, 61, 120, 112, 113funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( x
( 2nd `  G
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  G ) `  x
) (  Hom  `  E
) ( ( 1st `  G ) `  y
) ) )
146145, 115ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  G
) y ) `  f )  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  E ) ( ( 1st `  G
) `  y )
) )
1472, 3, 55, 111, 113, 114funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( y
( 2nd `  F
) z ) : ( y (  Hom  `  C ) z ) --> ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
148147, 116ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
y ( 2nd `  F
) z ) `  g )  e.  ( ( ( 1st `  F
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  z )
) )
1492, 3, 61, 120, 113, 114funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( y
( 2nd `  G
) z ) : ( y (  Hom  `  C ) z ) --> ( ( ( 1st `  G ) `  y
) (  Hom  `  E
) ( ( 1st `  G ) `  z
) ) )
150149, 116ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
y ( 2nd `  G
) z ) `  g )  e.  ( ( ( 1st `  G
) `  y )
(  Hom  `  E ) ( ( 1st `  G
) `  z )
) )
15116, 17, 18, 55, 61, 136, 138, 139, 140, 110, 118, 24, 141, 142, 144, 146, 148, 150xpcco2 14286 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( <. ( ( y ( 2nd `  F ) z ) `
 g ) ,  ( ( y ( 2nd `  G ) z ) `  g
) >. ( <. <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ,  <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. >. (comp `  T
) <. ( ( 1st `  F ) `  z
) ,  ( ( 1st `  G ) `
 z ) >.
) <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  <. (
( ( y ( 2nd `  F ) z ) `  g
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) ,  ( ( ( y ( 2nd `  G
) z ) `  g ) ( <.
( ( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  E
) ( ( 1st `  G ) `  z
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) >.
)
152134, 151eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
( y ( 2nd `  P ) z ) `
 g ) (
<. ( ( 1st `  P
) `  x ) ,  ( ( 1st `  P ) `  y
) >. (comp `  T
) ( ( 1st `  P ) `  z
) ) ( ( x ( 2nd `  P
) y ) `  f ) )  = 
<. ( ( ( y ( 2nd `  F
) z ) `  g ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) ,  ( ( ( y ( 2nd `  G
) z ) `  g ) ( <.
( ( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  E
) ( ( 1st `  G ) `  z
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) >.
)
153122, 126, 1523eqtr4d 2480 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) )  ->  ( (
x ( 2nd `  P
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  P ) z ) `
 g ) (
<. ( ( 1st `  P
) `  x ) ,  ( ( 1st `  P ) `  y
) >. (comp `  T
) ( ( 1st `  P ) `  z
) ) ( ( x ( 2nd `  P
) y ) `  f ) ) )
1542, 19, 3, 20, 21, 22, 23, 24, 27, 32, 48, 54, 89, 109, 153isfuncd 14064 . . 3  |-  ( ph  ->  ( 1st `  P
) ( C  Func  T ) ( 2nd `  P
) )
155 df-br 4215 . . 3  |-  ( ( 1st `  P ) ( C  Func  T
) ( 2nd `  P
)  <->  <. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
156154, 155sylib 190 . 2  |-  ( ph  -> 
<. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
15715, 156eqeltrd 2512 1  |-  ( ph  ->  P  e.  ( C 
Func  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   Rel wrel 4885    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891   Idccid 13892    Func cfunc 14053    X.c cxpc 14267   ⟨,⟩F cprf 14270
This theorem is referenced by:  prf1st  14303  prf2nd  14304  uncfcl  14334  uncf1  14335  uncf2  14336  yonedalem1  14371  yonedalem21  14372  yonedalem22  14377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-cat 13895  df-cid 13896  df-func 14057  df-xpc 14271  df-prf 14274
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