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Theorem prf1st 15334
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 2467 . . . . . . 7  |-  ( D  X.c  E )  =  ( D  X.c  E )
2 eqid 2467 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2467 . . . . . . . 8  |-  ( Base `  E )  =  (
Base `  E )
41, 2, 3xpcbas 15308 . . . . . . 7  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  ( D  X.c  E ) )
5 eqid 2467 . . . . . . 7  |-  ( Hom  `  ( D  X.c  E ) )  =  ( Hom  `  ( D  X.c  E ) )
6 prf1st.c . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 15093 . . . . . . . . . 10  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simprd 463 . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
109adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
11 prf1st.d . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
12 funcrcl 15093 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
1413simprd 463 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
16 eqid 2467 . . . . . . 7  |-  ( D  1stF  E )  =  ( D  1stF  E )
17 eqid 2467 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
18 relfunc 15092 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
19 1st2ndbr 6834 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 6, 19sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2117, 2, 20funcf1 15096 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 6022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
23 relfunc 15092 . . . . . . . . . . 11  |-  Rel  ( C  Func  E )
24 1st2ndbr 6834 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
2523, 11, 24sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
2617, 3, 25funcf1 15096 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
2726ffvelrnda 6022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
28 opelxpi 5031 . . . . . . . 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 661 . . . . . . 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 15322 . . . . . 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 5876 . . . . . . 7  |-  ( ( 1st `  F ) `
 x )  e. 
_V
32 fvex 5876 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  e. 
_V
3331, 32op1st 6793 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( ( 1st `  F ) `
 x )
3430, 33syl6eq 2524 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( ( 1st `  F
) `  x )
)
3534mpteq2dva 4533 . . . 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 2467 . . . . . . 7  |-  ( Hom  `  C )  =  ( Hom  `  C )
3836, 17, 37, 6, 11prfval 15329 . . . . . 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 5876 . . . . . . . 8  |-  ( Base `  C )  e.  _V
4039mptex 6132 . . . . . . 7  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
4139, 39mpt2ex 6861 . . . . . . 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 6795 . . . . . 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 15092 . . . . . . . 8  |-  Rel  (
( D  X.c  E ) 
Func  D )
451, 9, 14, 161stfcl 15327 . . . . . . . 8  |-  ( ph  ->  ( D  1stF  E )  e.  ( ( D  X.c  E
)  Func  D )
)
46 1st2ndbr 6834 . . . . . . . 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 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E ) 
Func  D ) ( 2nd `  ( D  1stF  E )
) )
484, 2, 47funcf1 15096 . . . . . 6  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) : ( ( Base `  D
)  X.  ( Base `  E ) ) --> (
Base `  D )
)
4948feqmptd 5921 . . . . 5  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  |->  ( ( 1st `  ( D  1stF  E ) ) `  u ) ) )
50 fveq2 5866 . . . . 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 6055 . . . 4  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( x  e.  ( Base `  C )  |->  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ) )
5221feqmptd 5921 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
5335, 51, 523eqtr4d 2518 . . 3  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( 1st `  F ) )
549ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  D  e.  Cat )
5514ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  E  e.  Cat )
56 relfunc 15092 . . . . . . . . . . . . . . . 16  |-  Rel  ( C  Func  ( D  X.c  E
) )
5736, 1, 6, 11prfcl 15333 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  ( C 
Func  ( D  X.c  E
) ) )
58 1st2ndbr 6834 . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  P
) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
6017, 4, 59funcf1 15096 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
6160ffvelrnda 6022 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6261adantrr 716 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6362adantr 465 . . . . . . . . . . 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 6022 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6564adantrl 715 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6665adantr 465 . . . . . . . . . . 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 15323 . . . . . . . . . 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 5868 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
70 simprl 755 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
71 simprr 756 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
7217, 37, 5, 69, 70, 71funcf2 15098 . . . . . . . . . . 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 6022 . . . . . . . . . 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 5880 . . . . . . . . . 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 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  F  e.  ( C  Func  D ) )
7711ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  G  e.  ( C  Func  E ) )
7870adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
7971adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
( Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
80 simpr 461 . . . . . . . . . . . 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 15332 . . . . . . . . . . 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 5870 . . . . . . . . . 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 5876 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) `  f )  e.  _V
84 fvex 5876 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  G
) y ) `  f )  e.  _V
8583, 84op1st 6793 . . . . . . . . . 10  |-  ( 1st `  <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  ( ( x ( 2nd `  F
) y ) `  f )
8682, 85syl6eq 2524 . . . . . . . . 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 2512 . . . . . . . 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 4533 . . . . . . 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 2467 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
9047adantr 465 . . . . . . . . 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 15098 . . . . . . . 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 6058 . . . . . . . 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 661 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
9517, 37, 89, 94, 70, 71funcf2 15098 . . . . . . . 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 5921 . . . . . . 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 2518 . . . . . 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 1192 . . . . 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 6346 . . . 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 15099 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
101 fnov 6395 . . . . 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 196 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
10399, 102eqtr4d 2511 . . 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 4221 . 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 15112 . 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 6831 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
10718, 6, 106sylancr 663 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
108104, 105, 1073eqtr4d 2518 1  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997    |` cres 5001    o. ccom 5003   Rel wrel 5004    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784   Basecbs 14493   Hom chom 14569   Catccat 14922    Func cfunc 15084    o.func ccofu 15086    X.c cxpc 15298    1stF c1stf 15299   ⟨,⟩F cprf 15301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-func 15088  df-cofu 15090  df-xpc 15302  df-1stf 15303  df-prf 15305
This theorem is referenced by: (None)
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