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Theorem prf1st 14332
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 2442 . . . . . . 7  |-  ( D  X.c  E )  =  ( D  X.c  E )
2 eqid 2442 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2442 . . . . . . . 8  |-  ( Base `  E )  =  (
Base `  E )
41, 2, 3xpcbas 14306 . . . . . . 7  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  ( D  X.c  E ) )
5 eqid 2442 . . . . . . 7  |-  (  Hom  `  ( D  X.c  E ) )  =  (  Hom  `  ( D  X.c  E ) )
6 prf1st.c . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 14091 . . . . . . . . . 10  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simprd 451 . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
109adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
11 prf1st.d . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
12 funcrcl 14091 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
1413simprd 451 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
1514adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
16 eqid 2442 . . . . . . 7  |-  ( D  1stF  E )  =  ( D  1stF  E )
17 eqid 2442 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
18 relfunc 14090 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
19 1st2ndbr 6425 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 6, 19sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2117, 2, 20funcf1 14094 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5899 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
23 relfunc 14090 . . . . . . . . . . 11  |-  Rel  ( C  Func  E )
24 1st2ndbr 6425 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
2523, 11, 24sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
2617, 3, 25funcf1 14094 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
2726ffvelrnda 5899 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
28 opelxpi 4939 . . . . . . . 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 644 . . . . . . 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 14320 . . . . . 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 5771 . . . . . . 7  |-  ( ( 1st `  F ) `
 x )  e. 
_V
32 fvex 5771 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  e. 
_V
3331, 32op1st 6384 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( ( 1st `  F ) `
 x )
3430, 33syl6eq 2490 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  1stF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( ( 1st `  F
) `  x )
)
3534mpteq2dva 4320 . . . 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 2442 . . . . . . 7  |-  (  Hom  `  C )  =  (  Hom  `  C )
3836, 17, 37, 6, 11prfval 14327 . . . . . 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 5771 . . . . . . . 8  |-  ( Base `  C )  e.  _V
4039mptex 5995 . . . . . . 7  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
4139, 39mpt2ex 6454 . . . . . . 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 6386 . . . . . 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 14090 . . . . . . . 8  |-  Rel  (
( D  X.c  E ) 
Func  D )
451, 9, 14, 161stfcl 14325 . . . . . . . 8  |-  ( ph  ->  ( D  1stF  E )  e.  ( ( D  X.c  E
)  Func  D )
)
46 1st2ndbr 6425 . . . . . . . 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 646 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) ( ( D  X.c  E ) 
Func  D ) ( 2nd `  ( D  1stF  E )
) )
484, 2, 47funcf1 14094 . . . . . 6  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) ) : ( ( Base `  D
)  X.  ( Base `  E ) ) --> (
Base `  D )
)
4948feqmptd 5808 . . . . 5  |-  ( ph  ->  ( 1st `  ( D  1stF  E ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  |->  ( ( 1st `  ( D  1stF  E ) ) `  u ) ) )
50 fveq2 5757 . . . . 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 5930 . . . 4  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( x  e.  ( Base `  C )  |->  ( ( 1st `  ( D  1stF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ) )
5221feqmptd 5808 . . . 4  |-  ( ph  ->  ( 1st `  F
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  F ) `  x
) ) )
5335, 51, 523eqtr4d 2484 . . 3  |-  ( ph  ->  ( ( 1st `  ( D  1stF  E ) )  o.  ( 1st `  P
) )  =  ( 1st `  F ) )
549ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
5514ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  E  e.  Cat )
56 relfunc 14090 . . . . . . . . . . . . . . . 16  |-  Rel  ( C  Func  ( D  X.c  E
) )
5736, 1, 6, 11prfcl 14331 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  ( C 
Func  ( D  X.c  E
) ) )
58 1st2ndbr 6425 . . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  P
) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
6017, 4, 59funcf1 14094 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
6160ffvelrnda 5899 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6261adantrr 699 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6362adantr 453 . . . . . . . . . . 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 5899 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6564adantrl 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6665adantr 453 . . . . . . . . . . 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 14321 . . . . . . . . . 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 5759 . . . . . . . . 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 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
70 simprl 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
71 simprr 735 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
7217, 37, 5, 69, 70, 71funcf2 14096 . . . . . . . . . . 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 5899 . . . . . . . . . 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 5774 . . . . . . . . . 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 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  F  e.  ( C  Func  D ) )
7711ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  G  e.  ( C  Func  E ) )
7870adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
7971adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
80 simpr 449 . . . . . . . . . . . 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 14330 . . . . . . . . . . 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 5761 . . . . . . . . . 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 5771 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) `  f )  e.  _V
84 fvex 5771 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  G
) y ) `  f )  e.  _V
8583, 84op1st 6384 . . . . . . . . . 10  |-  ( 1st `  <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  ( ( x ( 2nd `  F
) y ) `  f )
8682, 85syl6eq 2490 . . . . . . . . 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 2478 . . . . . . . 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 4320 . . . . . . 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 2442 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
9047adantr 453 . . . . . . . . 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 14096 . . . . . . . 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 5933 . . . . . . . 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 644 . . . . . . 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 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
9517, 37, 89, 94, 70, 71funcf2 14096 . . . . . . . 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 5808 . . . . . . 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 2484 . . . . . 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 1150 . . . . 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 6167 . . . 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 14097 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
101 fnov 6207 . . . . 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 190 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
10399, 102eqtr4d 2477 . . 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 4016 . 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 14110 . 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 6422 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
10718, 6, 106sylancr 646 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
108104, 105, 1073eqtr4d 2484 1  |-  ( ph  ->  ( ( D  1stF  E )  o.func 
P )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   <.cop 3841   class class class wbr 4237    e. cmpt 4291    X. cxp 4905    |` cres 4909    o. ccom 4911   Rel wrel 4912    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   1stc1st 6376   2ndc2nd 6377   Basecbs 13500    Hom chom 13571   Catccat 13920    Func cfunc 14082    o.func ccofu 14084    X.c cxpc 14296    1stF c1stf 14297   ⟨,⟩F cprf 14299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-hom 13584  df-cco 13585  df-cat 13924  df-cid 13925  df-func 14086  df-cofu 14088  df-xpc 14300  df-1stf 14301  df-prf 14303
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