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Theorem prf1st 14260
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 2408 . . . . . . 7  |-  ( D  X.c  E )  =  ( D  X.c  E )
2 eqid 2408 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2408 . . . . . . . 8  |-  ( Base `  E )  =  (
Base `  E )
41, 2, 3xpcbas 14234 . . . . . . 7  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  ( D  X.c  E ) )
5 eqid 2408 . . . . . . 7  |-  (  Hom  `  ( D  X.c  E ) )  =  (  Hom  `  ( D  X.c  E ) )
6 prf1st.c . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 14019 . . . . . . . . . 10  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simprd 450 . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
109adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
11 prf1st.d . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
12 funcrcl 14019 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
1413simprd 450 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
1514adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
16 eqid 2408 . . . . . . 7  |-  ( D  1stF  E )  =  ( D  1stF  E )
17 eqid 2408 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
18 relfunc 14018 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
19 1st2ndbr 6359 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 6, 19sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2117, 2, 20funcf1 14022 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5833 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
23 relfunc 14018 . . . . . . . . . . 11  |-  Rel  ( C  Func  E )
24 1st2ndbr 6359 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
2523, 11, 24sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
2617, 3, 25funcf1 14022 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
2726ffvelrnda 5833 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
28 opelxpi 4873 . . . . . . . 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 643 . . . . . . 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 14248 . . . . . 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 5705 . . . . . . 7  |-  ( ( 1st `  F ) `
 x )  e. 
_V
32 fvex 5705 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  e. 
_V
3331, 32op1st 6318 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( ( 1st `  F ) `
 x )
3430, 33syl6eq 2456 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( ( 1st `  F
) `  x )
)
3534mpteq2dva 4259 . . . 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 2408 . . . . . . 7  |-  (  Hom  `  C )  =  (  Hom  `  C )
3836, 17, 37, 6, 11prfval 14255 . . . . . 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 5705 . . . . . . . 8  |-  ( Base `  C )  e.  _V
4039mptex 5929 . . . . . . 7  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
4139, 39mpt2ex 6388 . . . . . . 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 6320 . . . . . 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 16 . . . . 5  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
44 relfunc 14018 . . . . . . . 8  |-  Rel  (
( D  X.c  E ) 
Func  D )
451, 9, 14, 161stfcl 14253 . . . . . . . 8  |-  ( ph  ->  ( D  1stF  E )  e.  ( ( D  X.c  E
)  Func  D )
)
46 1st2ndbr 6359 . . . . . . . 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 645 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E ) 
Func  D ) ( 2nd `  ( D  1stF  E )
) )
484, 2, 47funcf1 14022 . . . . . 6  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) : ( ( Base `  D
)  X.  ( Base `  E ) ) --> (
Base `  D )
)
4948feqmptd 5742 . . . . 5  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  |->  ( ( 1st `  ( D  1stF  E ) ) `  u ) ) )
50 fveq2 5691 . . . . 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 5864 . . . 4  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( x  e.  ( Base `  C )  |->  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ) )
5221feqmptd 5742 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
5335, 51, 523eqtr4d 2450 . . 3  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( 1st `  F ) )
549ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
5514ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  E  e.  Cat )
56 relfunc 14018 . . . . . . . . . . . . . . . 16  |-  Rel  ( C  Func  ( D  X.c  E
) )
5736, 1, 6, 11prfcl 14259 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  ( C 
Func  ( D  X.c  E
) ) )
58 1st2ndbr 6359 . . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  P
) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
6017, 4, 59funcf1 14022 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
6160ffvelrnda 5833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6261adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6362adantr 452 . . . . . . . . . . 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 5833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6564adantrl 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6665adantr 452 . . . . . . . . . . 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 14249 . . . . . . . . . 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 5693 . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
70 simprl 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
71 simprr 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
7217, 37, 5, 69, 70, 71funcf2 14024 . . . . . . . . . . 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 5833 . . . . . . . . . 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 5708 . . . . . . . . . 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 16 . . . . . . . . 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 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  F  e.  ( C  Func  D ) )
7711ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  G  e.  ( C  Func  E ) )
7870adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
7971adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
80 simpr 448 . . . . . . . . . . . 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 14258 . . . . . . . . . . 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 5695 . . . . . . . . . 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 5705 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) `  f )  e.  _V
84 fvex 5705 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  G
) y ) `  f )  e.  _V
8583, 84op1st 6318 . . . . . . . . . 10  |-  ( 1st `  <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  ( ( x ( 2nd `  F
) y ) `  f )
8682, 85syl6eq 2456 . . . . . . . . 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 2444 . . . . . . . 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 4259 . . . . . . 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 2408 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
9047adantr 452 . . . . . . . . 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 14024 . . . . . . . 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 5867 . . . . . . . 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 643 . . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
9517, 37, 89, 94, 70, 71funcf2 14024 . . . . . . . 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 5742 . . . . . . 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 2450 . . . . . 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 1149 . . . . 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 6101 . . . 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 14025 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
101 fnov 6141 . . . . 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 189 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
10399, 102eqtr4d 2443 . . 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 3956 . 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 14038 . 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 6356 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
10718, 6, 106sylancr 645 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
108104, 105, 1073eqtr4d 2450 1  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3781   class class class wbr 4176    e. cmpt 4230    X. cxp 4839    |` cres 4843    o. ccom 4845   Rel wrel 4846    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311   Basecbs 13428    Hom chom 13499   Catccat 13848    Func cfunc 14010    o.func ccofu 14012    X.c cxpc 14224    1stF c1stf 14225   ⟨,⟩F cprf 14227
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-hom 13512  df-cco 13513  df-cat 13852  df-cid 13853  df-func 14014  df-cofu 14016  df-xpc 14228  df-1stf 14229  df-prf 14231
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