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Theorem prfval 14251
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 14227 . . . 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 5701 . . . . . 6  |-  ( 1st `  f )  e.  _V
54dmex 5091 . . . . 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 5691 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
98dmeqd 5031 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  =  dom  ( 1st `  F ) )
10 prfval.b . . . . . . . 8  |-  B  =  ( Base `  C
)
11 eqid 2404 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
12 relfunc 14014 . . . . . . . . 9  |-  Rel  ( C  Func  D )
13 prfval.c . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 1st2ndbr 6355 . . . . . . . . 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 14018 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
17 fdm 5554 . . . . . . 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 2436 . . . 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 5691 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  f )  =  ( 1st `  F
) )
2423fveq1d 5689 . . . . . . 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 5691 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  g )  =  ( 1st `  G
) )
2726fveq1d 5689 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
( 1st `  g
) `  x )  =  ( ( 1st `  G ) `  x
) )
2824, 27opeq12d 3952 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  =  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
2921, 28mpteq12dv 4247 . . . . 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 2405 . . . . . . 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 6095 . . . . . 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 5691 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
3433oveqd 6057 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( x ( 2nd `  F ) y ) )
3534dmeqd 5031 . . . . . . . . . 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 2404 . . . . . . . . . . . 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 14020 . . . . . . . . . . 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 5554 . . . . . . . . . . 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 2436 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  f ) y )  =  ( x H y ) )
4534fveq1d 5689 . . . . . . . . . 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 5691 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
4847oveqd 6057 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  g
) y )  =  ( x ( 2nd `  G ) y ) )
4948fveq1d 5689 . . . . . . . . . 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 3952 . . . . . . . . 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 4247 . . . . . . . 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 6097 . . . . . 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 2436 . . . . 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 3952 . . . 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 3254 . . 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 2924 . . . 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 2924 . . . 4  |-  ( G  e.  ( C  Func  E )  ->  G  e.  _V )
6159, 60syl 16 . . 3  |-  ( ph  ->  G  e.  _V )
62 opex 4387 . . . 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 6160 . 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 2448 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 1721   _Vcvv 2916   [_csb 3211   <.cop 3777   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495    Func cfunc 14006   ⟨,⟩F cprf 14223
This theorem is referenced by:  prf1  14252  prf2fval  14253  prfcl  14255  prf1st  14256  prf2nd  14257  1st2ndprf  14258
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-ixp 7023  df-func 14010  df-prf 14227
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