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Theorem prfcl 13993
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 2296 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2296 . . . 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 13989 . . 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 5555 . . . . . . 7  |-  ( Base `  C )  e.  _V
87mptex 5762 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
97, 7mpt2ex 6214 . . . . . 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 6146 . . . . 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 15 . . . 4  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
128, 9op2ndd 6147 . . . . 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 15 . . . 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 3820 . . 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 2331 . 2  |-  ( ph  ->  P  =  <. ( 1st `  P ) ,  ( 2nd `  P
) >. )
16 prfcl.t . . . . 5  |-  T  =  ( D  X.c  E )
17 eqid 2296 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
18 eqid 2296 . . . . 5  |-  ( Base `  E )  =  (
Base `  E )
1916, 17, 18xpcbas 13968 . . . 4  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  T )
20 eqid 2296 . . . 4  |-  (  Hom  `  T )  =  (  Hom  `  T )
21 eqid 2296 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
22 eqid 2296 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
23 eqid 2296 . . . 4  |-  (comp `  C )  =  (comp `  C )
24 eqid 2296 . . . 4  |-  (comp `  T )  =  (comp `  T )
25 funcrcl 13753 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
264, 25syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2726simpld 445 . . . 4  |-  ( ph  ->  C  e.  Cat )
2826simprd 449 . . . . 5  |-  ( ph  ->  D  e.  Cat )
29 funcrcl 13753 . . . . . . 7  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
305, 29syl 15 . . . . . 6  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
3130simprd 449 . . . . 5  |-  ( ph  ->  E  e.  Cat )
3216, 28, 31xpccat 13980 . . . 4  |-  ( ph  ->  T  e.  Cat )
33 relfunc 13752 . . . . . . . . . 10  |-  Rel  ( C  Func  D )
34 1st2ndbr 6185 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3533, 4, 34sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
362, 17, 35funcf1 13756 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
3736ffvelrnda 5681 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
38 relfunc 13752 . . . . . . . . . 10  |-  Rel  ( C  Func  E )
39 1st2ndbr 6185 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
4038, 5, 39sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
412, 18, 40funcf1 13756 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
4241ffvelrnda 5681 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
43 opelxpi 4737 . . . . . . 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 642 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
45 eqid 2296 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4644, 45fmptd 5700 . . . . 5  |-  ( ph  ->  ( x  e.  (
Base `  C )  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) : (
Base `  C ) --> ( ( Base `  D
)  X.  ( Base `  E ) ) )
4711feq1d 5395 . . . . 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 223 . . . 4  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
49 eqid 2296 . . . . . 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 5899 . . . . . . 7  |-  ( x (  Hom  `  C
) y )  e. 
_V
5150mptex 5762 . . . . . 6  |-  ( h  e.  ( x (  Hom  `  C )
y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. )  e.  _V
5249, 51fnmpt2i 6209 . . . . 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 5351 . . . . 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 224 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
55 eqid 2296 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
5635adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
57 simprl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
58 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
592, 3, 55, 56, 57, 58funcf2 13758 . . . . . . . . 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 5681 . . . . . . . 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 2296 . . . . . . . . . 10  |-  (  Hom  `  E )  =  (  Hom  `  E )
6240adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
632, 3, 61, 62, 57, 58funcf2 13758 . . . . . . . . 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 5681 . . . . . . . 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 4737 . . . . . . . 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 642 . . . . . . 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 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C  Func  D
) )
685adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( C  Func  E
) )
691, 2, 3, 67, 68, 57prf1 13990 . . . . . . . . . 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 13990 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  =  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >.
)
7169, 70oveq12d 5892 . . . . . . . . 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 697 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
7342adantrr 697 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  E
) )
7436ffvelrnda 5681 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
7574adantrl 696 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
7641ffvelrnda 5681 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  E )
)
7776adantrl 696 . . . . . . . . . 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 13976 . . . . . . . . 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 2328 . . . . . . . 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 451 . . . . . . 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 2373 . . . . . 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 2296 . . . . . 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 5700 . . . . 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 5891 . . . . . . 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 5986 . . . . . . . 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 1266 . . . . . . 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 2348 . . . . . 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 5395 . . . . 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 223 . . . 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 2296 . . . . . . 7  |-  ( Id
`  D )  =  ( Id `  D
)
9135adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
92 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
932, 21, 90, 91, 92funcid 13760 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  D
) `  ( ( 1st `  F ) `  x ) ) )
94 eqid 2296 . . . . . . 7  |-  ( Id
`  E )  =  ( Id `  E
)
9540adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  G ) ( C 
Func  E ) ( 2nd `  G ) )
962, 21, 94, 95, 92funcid 13760 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  G
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  E
) `  ( ( 1st `  G ) `  x ) ) )
9793, 96opeq12d 3820 . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F  e.  ( C  Func  D ) )
995adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  G  e.  ( C  Func  E ) )
10027adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
1012, 3, 21, 100, 92catidcl 13600 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
1021, 2, 3, 98, 99, 92, 92, 101prf2 13992 . . . . 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 13990 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  =  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
104103fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  ( ( Id `  T ) `
 <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
10528adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
10631adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
10716, 105, 106, 17, 18, 90, 94, 22, 37, 42xpcid 13979 . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  <. (
( Id `  D
) `  ( ( 1st `  F ) `  x ) ) ,  ( ( Id `  E ) `  (
( 1st `  G
) `  x )
) >. )
10997, 102, 1083eqtr4d 2338 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  P
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  T
) `  ( ( 1st `  P ) `  x ) ) )
110 eqid 2296 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
111353ad2ant1 976 . . . . . . 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 988 . . . . . . 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 989 . . . . . . 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 990 . . . . . . 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 983 . . . . . . 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 984 . . . . . . 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 13761 . . . . . 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 2296 . . . . . . 7  |-  (comp `  E )  =  (comp `  E )
11953ad2ant1 976 . . . . . . . 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 644 . . . . . . 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 13761 . . . . . 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 3820 . . . . 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 976 . . . . . 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 976 . . . . . . 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 13603 . . . . . 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 13992 . . . . 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 13990 . . . . . . . . 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 13990 . . . . . . . . 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 3820 . . . . . . . 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 13990 . . . . . . . 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 5892 . . . . . . 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 13992 . . . . . . 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 13992 . . . . . . 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 5895 . . . . . 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 976 . . . . . . . 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 5682 . . . . . . 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 976 . . . . . . . 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 5682 . . . . . . 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 5682 . . . . . . 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 5682 . . . . . . 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 5682 . . . . . . 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 5682 . . . . . . 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 13758 . . . . . . . 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 5682 . . . . . . 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 13758 . . . . . . . 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 5682 . . . . . . 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 13758 . . . . . . . 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 5682 . . . . . . 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 13758 . . . . . . . 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 5682 . . . . . . 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 13977 . . . . . 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 2328 . . . . 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 2338 . . . 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 13755 . . 3  |-  ( ph  ->  ( 1st `  P
) ( C  Func  T ) ( 2nd `  P
) )
155 df-br 4040 . . 3  |-  ( ( 1st `  P ) ( C  Func  T
) ( 2nd `  P
)  <->  <. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
156154, 155sylib 188 . 2  |-  ( ph  -> 
<. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
15715, 156eqeltrd 2370 1  |-  ( ph  ->  P  e.  ( C 
Func  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583    Func cfunc 13744    X.c cxpc 13958   ⟨,⟩F cprf 13961
This theorem is referenced by:  prf1st  13994  prf2nd  13995  uncfcl  14025  uncf1  14026  uncf2  14027  yonedalem1  14062  yonedalem21  14063  yonedalem22  14068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-func 13748  df-xpc 13962  df-prf 13965
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