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Theorem prfcl 14255
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 2404 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2404 . . . 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 14251 . . 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 5701 . . . . . . 7  |-  ( Base `  C )  e.  _V
87mptex 5925 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
97, 7mpt2ex 6384 . . . . . 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 6316 . . . . 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 6317 . . . . 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 3952 . . 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 2439 . 2  |-  ( ph  ->  P  =  <. ( 1st `  P ) ,  ( 2nd `  P
) >. )
16 prfcl.t . . . . 5  |-  T  =  ( D  X.c  E )
17 eqid 2404 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
18 eqid 2404 . . . . 5  |-  ( Base `  E )  =  (
Base `  E )
1916, 17, 18xpcbas 14230 . . . 4  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  T )
20 eqid 2404 . . . 4  |-  (  Hom  `  T )  =  (  Hom  `  T )
21 eqid 2404 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
22 eqid 2404 . . . 4  |-  ( Id
`  T )  =  ( Id `  T
)
23 eqid 2404 . . . 4  |-  (comp `  C )  =  (comp `  C )
24 eqid 2404 . . . 4  |-  (comp `  T )  =  (comp `  T )
25 funcrcl 14015 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
264, 25syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2726simpld 446 . . . 4  |-  ( ph  ->  C  e.  Cat )
2826simprd 450 . . . . 5  |-  ( ph  ->  D  e.  Cat )
29 funcrcl 14015 . . . . . . 7  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
305, 29syl 16 . . . . . 6  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
3130simprd 450 . . . . 5  |-  ( ph  ->  E  e.  Cat )
3216, 28, 31xpccat 14242 . . . 4  |-  ( ph  ->  T  e.  Cat )
33 relfunc 14014 . . . . . . . . . 10  |-  Rel  ( C  Func  D )
34 1st2ndbr 6355 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3533, 4, 34sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
362, 17, 35funcf1 14018 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
3736ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
38 relfunc 14014 . . . . . . . . . 10  |-  Rel  ( C  Func  E )
39 1st2ndbr 6355 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
4038, 5, 39sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
412, 18, 40funcf1 14018 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
4241ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
43 opelxpi 4869 . . . . . . 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 643 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
45 eqid 2404 . . . . . 6  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)
4644, 45fmptd 5852 . . . . 5  |-  ( ph  ->  ( x  e.  (
Base `  C )  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) : (
Base `  C ) --> ( ( Base `  D
)  X.  ( Base `  E ) ) )
4711feq1d 5539 . . . . 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 224 . . . 4  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
49 eqid 2404 . . . . . 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 6065 . . . . . . 7  |-  ( x (  Hom  `  C
) y )  e. 
_V
5150mptex 5925 . . . . . 6  |-  ( h  e.  ( x (  Hom  `  C )
y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. )  e.  _V
5249, 51fnmpt2i 6379 . . . . 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 5495 . . . . 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 225 . . . 4  |-  ( ph  ->  ( 2nd `  P
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
55 eqid 2404 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
5635adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
57 simprl 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
58 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
592, 3, 55, 56, 57, 58funcf2 14020 . . . . . . . . 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 5829 . . . . . . . 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 2404 . . . . . . . . . 10  |-  (  Hom  `  E )  =  (  Hom  `  E )
6240adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
632, 3, 61, 62, 57, 58funcf2 14020 . . . . . . . . 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 5829 . . . . . . . 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 4869 . . . . . . . 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 643 . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C  Func  D
) )
685adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( C  Func  E
) )
691, 2, 3, 67, 68, 57prf1 14252 . . . . . . . . . 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 14252 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  =  <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 y ) >.
)
7169, 70oveq12d 6058 . . . . . . . . 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 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
7342adantrr 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  E
) )
7436ffvelrnda 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
7574adantrl 697 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
7641ffvelrnda 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  E )
)
7776adantrl 697 . . . . . . . . . 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 14238 . . . . . . . . 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 2436 . . . . . . . 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 452 . . . . . . 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 2481 . . . . . 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 2404 . . . . . 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 5852 . . . . 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 6057 . . . . . . 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 6155 . . . . . . . 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 1268 . . . . . . 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 2456 . . . . . 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 5539 . . . . 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 224 . . . 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 2404 . . . . . . 7  |-  ( Id
`  D )  =  ( Id `  D
)
9135adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
92 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
932, 21, 90, 91, 92funcid 14022 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  D
) `  ( ( 1st `  F ) `  x ) ) )
94 eqid 2404 . . . . . . 7  |-  ( Id
`  E )  =  ( Id `  E
)
9540adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  G ) ( C 
Func  E ) ( 2nd `  G ) )
962, 21, 94, 95, 92funcid 14022 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  G
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  E
) `  ( ( 1st `  G ) `  x ) ) )
9793, 96opeq12d 3952 . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F  e.  ( C  Func  D ) )
995adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  G  e.  ( C  Func  E ) )
10027adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
1012, 3, 21, 100, 92catidcl 13862 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
1021, 2, 3, 98, 99, 92, 92, 101prf2 14254 . . . . 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 14252 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  =  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
104103fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  ( ( Id `  T ) `
 <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
10528adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
10631adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
10716, 105, 106, 17, 18, 90, 94, 22, 37, 42xpcid 14241 . . . . . 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 2436 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  T ) `  ( ( 1st `  P
) `  x )
)  =  <. (
( Id `  D
) `  ( ( 1st `  F ) `  x ) ) ,  ( ( Id `  E ) `  (
( 1st `  G
) `  x )
) >. )
10997, 102, 1083eqtr4d 2446 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  P
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  T
) `  ( ( 1st `  P ) `  x ) ) )
110 eqid 2404 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
111353ad2ant1 978 . . . . . . 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 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 ) ) )  ->  x  e.  ( Base `  C )
)
113 simp22 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 ) ) )  ->  y  e.  ( Base `  C )
)
114 simp23 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 ) ) )  ->  z  e.  ( Base `  C )
)
115 simp3l 985 . . . . . . 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 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 ) ) )  ->  g  e.  ( y (  Hom  `  C ) z ) )
1172, 3, 23, 110, 111, 112, 113, 114, 115, 116funcco 14023 . . . . . 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 2404 . . . . . . 7  |-  (comp `  E )  =  (comp `  E )
11953ad2ant1 978 . . . . . . . 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 645 . . . . . . 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 14023 . . . . . 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 3952 . . . . 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 978 . . . . . 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 978 . . . . . . 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 13865 . . . . . 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 14254 . . . . 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 14252 . . . . . . . . 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 14252 . . . . . . . . 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 3952 . . . . . . . 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 14252 . . . . . . . 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 6058 . . . . . . 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 14254 . . . . . . 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 14254 . . . . . . 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 6061 . . . . . 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 978 . . . . . . . 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 5830 . . . . . . 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 978 . . . . . . . 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 5830 . . . . . . 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 5830 . . . . . . 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 5830 . . . . . . 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 5830 . . . . . . 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 5830 . . . . . . 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 14020 . . . . . . . 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 5830 . . . . . . 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 14020 . . . . . . . 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 5830 . . . . . . 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 14020 . . . . . . . 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 5830 . . . . . . 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 14020 . . . . . . . 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 5830 . . . . . . 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 14239 . . . . . 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 2436 . . . . 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 2446 . . . 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 14017 . . 3  |-  ( ph  ->  ( 1st `  P
) ( C  Func  T ) ( 2nd `  P
) )
155 df-br 4173 . . 3  |-  ( ( 1st `  P ) ( C  Func  T
) ( 2nd `  P
)  <->  <. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
156154, 155sylib 189 . 2  |-  ( ph  -> 
<. ( 1st `  P
) ,  ( 2nd `  P ) >.  e.  ( C  Func  T )
)
15715, 156eqeltrd 2478 1  |-  ( ph  ->  P  e.  ( C 
Func  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   Rel wrel 4842    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845    Func cfunc 14006    X.c cxpc 14220   ⟨,⟩F cprf 14223
This theorem is referenced by:  prf1st  14256  prf2nd  14257  uncfcl  14287  uncf1  14288  uncf2  14289  yonedalem1  14324  yonedalem21  14325  yonedalem22  14330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-func 14010  df-xpc 14224  df-prf 14227
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