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Theorem prfval 13989
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 13965 . . . 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 10 . . 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 5555 . . . . . 6  |-  ( 1st `  f )  e.  _V
54dmex 4957 . . . . 5  |-  dom  ( 1st `  f )  e. 
_V
65a1i 10 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  e.  _V )
7 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
f  =  F )
87fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
98dmeqd 4897 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  =  dom  ( 1st `  F ) )
10 prfval.b . . . . . . . 8  |-  B  =  ( Base `  C
)
11 eqid 2296 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
12 relfunc 13752 . . . . . . . . 9  |-  Rel  ( C  Func  D )
13 prfval.c . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 1st2ndbr 6185 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1512, 13, 14sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1610, 11, 15funcf1 13756 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
17 fdm 5409 . . . . . . 7  |-  ( ( 1st `  F ) : B --> ( Base `  D )  ->  dom  ( 1st `  F )  =  B )
1816, 17syl 15 . . . . . 6  |-  ( ph  ->  dom  ( 1st `  F
)  =  B )
1918adantr 451 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  F
)  =  B )
209, 19eqtrd 2328 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  dom  ( 1st `  f
)  =  B )
21 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  b  =  B )
22 simplrl 736 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  f  =  F )
2322fveq2d 5545 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  f )  =  ( 1st `  F
) )
2423fveq1d 5543 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
25 simplrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  g  =  G )
2625fveq2d 5545 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  ( 1st `  g )  =  ( 1st `  G
) )
2726fveq1d 5543 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  (
( 1st `  g
) `  x )  =  ( ( 1st `  G ) `  x
) )
2824, 27opeq12d 3820 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  b  =  B )  ->  <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  =  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )
2921, 28mpteq12dv 4114 . . . . 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 2297 . . . . . . 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 5926 . . . . . 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 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  f  =  F )
3332fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
3433oveqd 5891 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( x ( 2nd `  F ) y ) )
3534dmeqd 4897 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  (  Hom  `  D )  =  (  Hom  `  D )
3815ad4antr 712 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
39 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  x  e.  B )
40 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  y  e.  B )
4110, 36, 37, 38, 39, 40funcf2 13758 . . . . . . . . . . 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 5409 . . . . . . . . . . 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 15 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  F ) y )  =  ( x H y ) )
4435, 43eqtrd 2328 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  dom  ( x ( 2nd `  f ) y )  =  ( x H y ) )
4534fveq1d 5543 . . . . . . . . . 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 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  g  =  G )
4746fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
4847oveqd 5891 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  =  F  /\  g  =  G ) )  /\  b  =  B )  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( 2nd `  g
) y )  =  ( x ( 2nd `  G ) y ) )
4948fveq1d 5543 . . . . . . . . . 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 3820 . . . . . . . . 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 4114 . . . . . . . 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 1146 . . . . . . 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 5928 . . . . . 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 2328 . . . . 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 3820 . . . 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 3137 . . 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 2809 . . . 4  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
5813, 57syl 15 . . 3  |-  ( ph  ->  F  e.  _V )
59 prfval.d . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
60 elex 2809 . . . 4  |-  ( G  e.  ( C  Func  E )  ->  G  e.  _V )
6159, 60syl 15 . . 3  |-  ( ph  ->  G  e.  _V )
62 opex 4253 . . . 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 10 . . 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 5991 . 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 2340 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   <.cop 3656   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235    Func cfunc 13744   ⟨,⟩F cprf 13961
This theorem is referenced by:  prf1  13990  prf2fval  13991  prfcl  13993  prf1st  13994  prf2nd  13995  1st2ndprf  13996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-func 13748  df-prf 13965
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