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Theorem prf1st 13978
Description: Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prf1st.p  |-  P  =  ( F ⟨,⟩F  G )
prf1st.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prf1st.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
Assertion
Ref Expression
prf1st  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  F )

Proof of Theorem prf1st
Dummy variables  f  h  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . . 7  |-  ( D  X.c  E )  =  ( D  X.c  E )
2 eqid 2283 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2283 . . . . . . . 8  |-  ( Base `  E )  =  (
Base `  E )
41, 2, 3xpcbas 13952 . . . . . . 7  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  ( D  X.c  E ) )
5 eqid 2283 . . . . . . 7  |-  (  Hom  `  ( D  X.c  E ) )  =  (  Hom  `  ( D  X.c  E ) )
6 prf1st.c . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 13737 . . . . . . . . . 10  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 15 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simprd 449 . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
11 prf1st.d . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
12 funcrcl 13737 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 15 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
1413simprd 449 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
1514adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
16 eqid 2283 . . . . . . 7  |-  ( D  1stF  E )  =  ( D  1stF  E )
17 eqid 2283 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
18 relfunc 13736 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
19 1st2ndbr 6169 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 6, 19sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2117, 2, 20funcf1 13740 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5665 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
23 relfunc 13736 . . . . . . . . . . 11  |-  Rel  ( C  Func  E )
24 1st2ndbr 6169 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
2523, 11, 24sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
2617, 3, 25funcf1 13740 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
2726ffvelrnda 5665 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
28 opelxpi 4721 . . . . . . . 8  |-  ( ( ( ( 1st `  F
) `  x )  e.  ( Base `  D
)  /\  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
2922, 27, 28syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
301, 4, 5, 10, 15, 16, 291stf1 13966 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( 1st `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
31 fvex 5539 . . . . . . 7  |-  ( ( 1st `  F ) `
 x )  e. 
_V
32 fvex 5539 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  e. 
_V
3331, 32op1st 6128 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( ( 1st `  F ) `
 x )
3430, 33syl6eq 2331 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( ( 1st `  F
) `  x )
)
3534mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )  =  ( x  e.  (
Base `  C )  |->  ( ( 1st `  F
) `  x )
) )
36 prf1st.p . . . . . . 7  |-  P  =  ( F ⟨,⟩F  G )
37 eqid 2283 . . . . . . 7  |-  (  Hom  `  C )  =  (  Hom  `  C )
3836, 17, 37, 6, 11prfval 13973 . . . . . 6  |-  ( 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 ) >.
) ) >. )
39 fvex 5539 . . . . . . . 8  |-  ( Base `  C )  e.  _V
4039mptex 5746 . . . . . . 7  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
4139, 39mpt2ex 6198 . . . . . . 7  |-  ( 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
4240, 41op1std 6130 . . . . . 6  |-  ( 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
) >. ) )
4338, 42syl 15 . . . . 5  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
44 relfunc 13736 . . . . . . . 8  |-  Rel  (
( D  X.c  E ) 
Func  D )
451, 9, 14, 161stfcl 13971 . . . . . . . 8  |-  ( ph  ->  ( D  1stF  E )  e.  ( ( D  X.c  E
)  Func  D )
)
46 1st2ndbr 6169 . . . . . . . 8  |-  ( ( Rel  ( ( D  X.c  E )  Func  D
)  /\  ( D  1stF  E )  e.  ( ( D  X.c  E )  Func  D
) )  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E )  Func  D
) ( 2nd `  ( D  1stF  E ) ) )
4744, 45, 46sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E ) 
Func  D ) ( 2nd `  ( D  1stF  E )
) )
484, 2, 47funcf1 13740 . . . . . 6  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) : ( ( Base `  D
)  X.  ( Base `  E ) ) --> (
Base `  D )
)
4948feqmptd 5575 . . . . 5  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  |->  ( ( 1st `  ( D  1stF  E ) ) `  u ) ) )
50 fveq2 5525 . . . . 5  |-  ( u  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  ->  ( ( 1st `  ( D  1stF  E ) ) `  u )  =  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
5129, 43, 49, 50fmptco 5691 . . . 4  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( x  e.  ( Base `  C )  |->  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ) )
5221feqmptd 5575 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
5335, 51, 523eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( 1st `  F ) )
549ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
5514ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  E  e.  Cat )
56 relfunc 13736 . . . . . . . . . . . . . . . 16  |-  Rel  ( C  Func  ( D  X.c  E
) )
5736, 1, 6, 11prfcl 13977 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  ( C 
Func  ( D  X.c  E
) ) )
58 1st2ndbr 6169 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( C  Func  ( D  X.c  E ) )  /\  P  e.  ( C  Func  ( D  X.c  E ) ) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
5956, 57, 58sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  P
) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
6017, 4, 59funcf1 13740 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
6160ffvelrnda 5665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6261adantrr 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6362adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6460ffvelrnda 5665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6564adantrl 696 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6665adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
671, 4, 5, 54, 55, 16, 63, 661stf2 13967 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  =  ( 1st  |`  ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) )
6867fveq1d 5527 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) )  =  ( ( 1st  |`  (
( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
6959adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
70 simprl 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
71 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
7217, 37, 5, 69, 70, 71funcf2 13742 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  P
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )
7372ffvelrnda 5665 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  P ) y ) `  f
)  e.  ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )
74 fvres 5542 . . . . . . . . . 10  |-  ( ( ( x ( 2nd `  P ) y ) `
 f )  e.  ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
)  ->  ( ( 1st  |`  ( ( ( 1st `  P ) `
 x ) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 1st `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
7573, 74syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 1st  |`  (
( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 1st `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
766ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  F  e.  ( C  Func  D ) )
7711ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  G  e.  ( C  Func  E ) )
7870adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
7971adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
80 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
f  e.  ( x (  Hom  `  C
) y ) )
8136, 17, 37, 76, 77, 78, 79, 80prf2 13976 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  P ) y ) `  f
)  =  <. (
( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. )
8281fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 1st `  <. ( ( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. ) )
83 fvex 5539 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) `  f )  e.  _V
84 fvex 5539 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  G
) y ) `  f )  e.  _V
8583, 84op1st 6128 . . . . . . . . . 10  |-  ( 1st `  <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  ( ( x ( 2nd `  F
) y ) `  f )
8682, 85syl6eq 2331 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( ( x ( 2nd `  F
) y ) `  f ) )
8768, 75, 863eqtrd 2319 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) )  =  ( ( x ( 2nd `  F ) y ) `  f
) )
8887mpteq2dva 4106 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
f  e.  ( x (  Hom  `  C
) y )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( x ( 2nd `  F ) y ) `  f
) ) )
89 eqid 2283 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
9047adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E )  Func  D
) ( 2nd `  ( D  1stF  E ) ) )
914, 5, 89, 90, 62, 65funcf2 13742 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  P
) `  x )
( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) : ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) --> ( ( ( 1st `  ( D  1stF  E ) ) `  ( ( 1st `  P
) `  x )
) (  Hom  `  D
) ( ( 1st `  ( D  1stF  E )
) `  ( ( 1st `  P ) `  y ) ) ) )
92 fcompt 5694 . . . . . . . 8  |-  ( ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) : ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) --> ( ( ( 1st `  ( D  1stF  E ) ) `  ( ( 1st `  P
) `  x )
) (  Hom  `  D
) ( ( 1st `  ( D  1stF  E )
) `  ( ( 1st `  P ) `  y ) ) )  /\  ( x ( 2nd `  P ) y ) : ( x (  Hom  `  C
) y ) --> ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) ) )
9391, 72, 92syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) ) )
9420adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
9517, 37, 89, 94, 70, 71funcf2 13742 . . . . . . . 8  |-  ( (
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
) ) )
9695feqmptd 5575 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y )  =  ( f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  F ) y ) `
 f ) ) )
9788, 93, 963eqtr4d 2325 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( x ( 2nd `  F ) y ) )
98973impb 1147 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( x ( 2nd `  F ) y ) )
9998mpt2eq3dva 5912 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
10017, 20funcfn2 13743 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
101 fnov 5952 . . . . 5  |-  ( ( 2nd `  F )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
102100, 101sylib 188 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
10399, 102eqtr4d 2318 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) )  =  ( 2nd `  F ) )
10453, 103opeq12d 3804 . 2  |-  ( ph  -> 
<. ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) ) >.  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
10517, 57, 45cofuval 13756 . 2  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  <. ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  1stF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) ) >. )
106 1st2nd 6166 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
10718, 6, 106sylancr 644 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
108104, 105, 1073eqtr4d 2325 1  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    |` cres 4691    o. ccom 4693   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566    Func cfunc 13728    o.func ccofu 13730    X.c cxpc 13942    1stF c1stf 13943   ⟨,⟩F cprf 13945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-func 13732  df-cofu 13734  df-xpc 13946  df-1stf 13947  df-prf 13949
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