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Theorem prfval 14223
Description: Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfval.k  |-  P  =  ( F ⟨,⟩F  G )
prfval.b  |-  B  =  ( Base `  C
)
prfval.h  |-  H  =  (  Hom  `  C
)
prfval.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prfval.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
Assertion
Ref Expression
prfval  |-  ( ph  ->  P  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
Distinct variable groups:    x, h, y, B    x, C, y   
h, F, x, y    ph, h, x, y    x, D, y    h, G, x, y    h, H, x, y
Allowed substitution hints:    C( h)    D( h)    P( x, y, h)    E( x, y, h)

Proof of Theorem prfval
Dummy variables  f 
b  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfval.k . 2  |-  P  =  ( F ⟨,⟩F  G )
2 df-prf 14199 . . . 4  |- ⟨,⟩F  =  ( f  e. 
_V ,  g  e. 
_V  |->  [_ dom  ( 1st `  f )  /  b ]_ <. ( x  e.  b  |->  <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
) ,  ( x  e.  b ,  y  e.  b  |->  ( h  e.  dom  ( x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
) >. )
32a1i 11 . . 3  |-  ( ph  -> ⟨,⟩F  =  ( f  e.  _V ,  g  e.  _V  |->  [_
dom  ( 1st `  f
)  /  b ]_ <. ( x  e.  b 
|->  <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
) ,  ( x  e.  b ,  y  e.  b  |->  ( h  e.  dom  ( x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
) >. ) )
4 fvex 5682 . . . . . 6  |-  ( 1st `  f )  e.  _V
54dmex 5072 . . . . 5  |-  dom  ( 1st `  f )  e. 
_V
65a1i 11 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  e.  _V )
7 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
f  =  F )
87fveq2d 5672 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
98dmeqd 5012 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  =  dom  ( 1st `  F ) )
10 prfval.b . . . . . . . 8  |-  B  =  ( Base `  C
)
11 eqid 2387 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
12 relfunc 13986 . . . . . . . . 9  |-  Rel  ( C  Func  D )
13 prfval.c . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 1st2ndbr 6335 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1512, 13, 14sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1610, 11, 15funcf1 13990 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
17 fdm 5535 . . . . . . 7  |-  ( ( 1st `  F ) : B --> ( Base `  D )  ->  dom  ( 1st `  F )  =  B )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  dom  ( 1st `  F
)  =  B )
1918adantr 452 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  F
)  =  B )
209, 19eqtrd 2419 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  =  B )
21 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  b  =  B )
22 simplrl 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  f  =  F )
2322fveq2d 5672 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  f )  =  ( 1st `  F
) )
2423fveq1d 5670 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
25 simplrr 738 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  g  =  G )
2625fveq2d 5672 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  g )  =  ( 1st `  G
) )
2726fveq1d 5670 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
( 1st `  g
) `  x )  =  ( ( 1st `  G ) `  x
) )
2824, 27opeq12d 3934 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  =  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
2921, 28mpteq12dv 4228 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
x  e.  b  |->  <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. )  =  ( x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
30 eqidd 2388 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )  =  ( h  e. 
dom  ( x ( 2nd `  f ) y )  |->  <. (
( x ( 2nd `  f ) y ) `
 h ) ,  ( ( x ( 2nd `  g ) y ) `  h
) >. ) )
3121, 21, 30mpt2eq123dv 6075 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  dom  ( x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
) )
3222ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  f  =  F )
3332fveq2d 5672 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
3433oveqd 6037 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( x ( 2nd `  F ) y ) )
3534dmeqd 5012 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  f ) y )  =  dom  ( x ( 2nd `  F
) y ) )
36 prfval.h . . . . . . . . . . . 12  |-  H  =  (  Hom  `  C
)
37 eqid 2387 . . . . . . . . . . . 12  |-  (  Hom  `  D )  =  (  Hom  `  D )
3815ad4antr 713 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
39 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  x  e.  B )
40 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  y  e.  B )
4110, 36, 37, 38, 39, 40funcf2 13992 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  F
) y ) : ( x H y ) --> ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  F
) `  y )
) )
42 fdm 5535 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) : ( x H y ) --> ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  F
) `  y )
)  ->  dom  ( x ( 2nd `  F
) y )  =  ( x H y ) )
4341, 42syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  F ) y )  =  ( x H y ) )
4435, 43eqtrd 2419 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  f ) y )  =  ( x H y ) )
4534fveq1d 5670 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
( x ( 2nd `  f ) y ) `
 h )  =  ( ( x ( 2nd `  F ) y ) `  h
) )
4625ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  g  =  G )
4746fveq2d 5672 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
4847oveqd 6037 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  g
) y )  =  ( x ( 2nd `  G ) y ) )
4948fveq1d 5670 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
( x ( 2nd `  g ) y ) `
 h )  =  ( ( x ( 2nd `  G ) y ) `  h
) )
5045, 49opeq12d 3934 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  <. (
( x ( 2nd `  f ) y ) `
 h ) ,  ( ( x ( 2nd `  g ) y ) `  h
) >.  =  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. )
5144, 50mpteq12dv 4228 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )  =  ( h  e.  ( x H y )  |->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )
52513impa 1148 . . . . . . 7  |-  ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B  /\  y  e.  B )  ->  (
h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )  =  ( h  e.  ( x H y )  |->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )
5352mpt2eq3dva 6077 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
x  e.  B , 
y  e.  B  |->  ( h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
5431, 53eqtrd 2419 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
5529, 54opeq12d 3934 . . . 4  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  <. (
x  e.  b  |->  <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. ) ,  ( x  e.  b ,  y  e.  b  |->  ( h  e.  dom  (
x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
) >.  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
566, 20, 55csbied2 3237 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  [_ dom  ( 1st `  f
)  /  b ]_ <. ( x  e.  b 
|->  <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
) ,  ( x  e.  b ,  y  e.  b  |->  ( h  e.  dom  ( x ( 2nd `  f
) y )  |->  <.
( ( x ( 2nd `  f ) y ) `  h
) ,  ( ( x ( 2nd `  g
) y ) `  h ) >. )
) >.  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
57 elex 2907 . . . 4  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
5813, 57syl 16 . . 3  |-  ( ph  ->  F  e.  _V )
59 prfval.d . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
60 elex 2907 . . . 4  |-  ( G  e.  ( C  Func  E )  ->  G  e.  _V )
6159, 60syl 16 . . 3  |-  ( ph  ->  G  e.  _V )
62 opex 4368 . . . 4  |-  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >.  e.  _V
6362a1i 11 . . 3  |-  ( ph  -> 
<. ( x  e.  B  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >.  e.  _V )
643, 56, 58, 61, 63ovmpt2d 6140 . 2  |-  ( ph  ->  ( F ⟨,⟩F  G )  =  <. ( x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
651, 64syl5eq 2431 1  |-  ( ph  ->  P  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   [_csb 3194   <.cop 3760   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   Rel wrel 4823   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287   Basecbs 13396    Hom chom 13467    Func cfunc 13978   ⟨,⟩F cprf 14195
This theorem is referenced by:  prf1  14224  prf2fval  14225  prfcl  14227  prf1st  14228  prf2nd  14229  1st2ndprf  14230
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956  df-ixp 7000  df-func 13982  df-prf 14199
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