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Theorem wwlktovfo 13012
Description: Lemma 3 for wrd2f1tovbij 13014. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wrd2f1tovbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
wrd2f1tovbij.r  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
wrd2f1tovbij.f  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
Assertion
Ref Expression
wwlktovfo  |-  ( P  e.  V  ->  F : D -onto-> R )
Distinct variable groups:    t, D    P, n, t, w    t, R    n, V, t, w   
n, X, w
Allowed substitution hints:    D( w, n)    R( w, n)    F( w, t, n)    X( t)

Proof of Theorem wwlktovfo
Dummy variables  p  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd2f1tovbij.d . . . 4  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
2 wrd2f1tovbij.r . . . 4  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
3 wrd2f1tovbij.f . . . 4  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
41, 2, 3wwlktovf 13010 . . 3  |-  F : D
--> R
54a1i 11 . 2  |-  ( P  e.  V  ->  F : D --> R )
6 preq2 4083 . . . . . 6  |-  ( n  =  p  ->  { P ,  n }  =  { P ,  p }
)
76eleq1d 2498 . . . . 5  |-  ( n  =  p  ->  ( { P ,  n }  e.  X  <->  { P ,  p }  e.  X )
)
87, 2elrab2 3237 . . . 4  |-  ( p  e.  R  <->  ( p  e.  V  /\  { P ,  p }  e.  X
) )
9 simpl 458 . . . . . . . . . . 11  |-  ( ( p  e.  V  /\  { P ,  p }  e.  X )  ->  p  e.  V )
109anim2i 571 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( P  e.  V  /\  p  e.  V ) )
11 eqidd 2430 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  { <. 0 ,  P >. , 
<. 1 ,  p >. } )
12 wrdlen2i 13000 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  p  e.  V )  ->  ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  { <. 0 ,  P >. , 
<. 1 ,  p >. }  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) ) )
1310, 11, 12sylc 62 . . . . . . . . 9  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )
14 prex 4664 . . . . . . . . . . 11  |-  { <. 0 ,  P >. , 
<. 1 ,  p >. }  e.  _V
1514a1i 11 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  { <. 0 ,  P >. ,  <. 1 ,  p >. }  e.  _V )
16 eleq1 2501 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  <->  u  e. Word  V ) )
1716biimpd 210 . . . . . . . . . . . . . . . . . . 19  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  ->  u  e. Word  V ) )
1817adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  ->  u  e. Word  V )
)
1918com12 32 . . . . . . . . . . . . . . . . 17  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  u  e. Word  V ) )
2019adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  u  e. Word  V ) )
2120adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  u  e. Word  V ) )
2221impcom 431 . . . . . . . . . . . . . 14  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  u  e. Word  V )
23 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  (
# `  u )
)
2423eqeq1d 2431 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  <-> 
( # `  u )  =  2 ) )
2524biimpd 210 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  ->  ( # `  u
)  =  2 ) )
2625adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  ->  ( # `  u
)  =  2 ) )
2726com12 32 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  { <. 0 ,  P >. ,  <. 1 ,  p >. } )  =  2  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  ( # `
 u )  =  2 ) )
2827adantl 467 . . . . . . . . . . . . . . . . 17  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  ( # `
 u )  =  2 ) )
2928adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  ( # `
 u )  =  2 ) )
3029impcom 431 . . . . . . . . . . . . . . 15  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  ( # `  u
)  =  2 )
31 fveq1 5880 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0
)  =  ( u `
 0 ) )
3231eqeq1d 2431 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  0 )  =  P  <->  ( u `  0 )  =  P ) )
3332biimpd 210 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  0 )  =  P  ->  (
u `  0 )  =  P ) )
3433adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0
)  =  P  -> 
( u `  0
)  =  P ) )
3534com12 32 . . . . . . . . . . . . . . . . . 18  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  ->  (
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  ->  ( u ` 
0 )  =  P ) )
3635adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0
)  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p )  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
u `  0 )  =  P ) )
3736adantl 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
u `  0 )  =  P ) )
3837impcom 431 . . . . . . . . . . . . . . 15  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  ( u `  0 )  =  P )
39 fveq1 5880 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1
)  =  ( u `
 1 ) )
4039eqeq1d 2431 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  1 )  =  p  <->  ( u `  1 )  =  p ) )
4132, 40anbi12d 715 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p )  <->  ( (
u `  0 )  =  P  /\  (
u `  1 )  =  p ) ) )
42 preq12 4084 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  { ( u `
 0 ) ,  ( u `  1
) }  =  { P ,  p }
)
4342eqcomd 2437 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  { P ,  p }  =  {
( u `  0
) ,  ( u `
 1 ) } )
4443eleq1d 2498 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  ( { P ,  p }  e.  X  <->  { ( u `  0
) ,  ( u `
 1 ) }  e.  X ) )
4544biimpd 210 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  ( { P ,  p }  e.  X  ->  { ( u ` 
0 ) ,  ( u `  1 ) }  e.  X ) )
4641, 45syl6bi 231 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p )  -> 
( { P ,  p }  e.  X  ->  { ( u ` 
0 ) ,  ( u `  1 ) }  e.  X ) ) )
4746com12 32 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0
)  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p )  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  ->  ( { P ,  p }  e.  X  ->  { ( u ` 
0 ) ,  ( u `  1 ) }  e.  X ) ) )
4847adantl 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  =  u  -> 
( { P ,  p }  e.  X  ->  { ( u ` 
0 ) ,  ( u `  1 ) }  e.  X ) ) )
4948com13 83 . . . . . . . . . . . . . . . . . 18  |-  ( { P ,  p }  e.  X  ->  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) ) )
5049ad2antll 733 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) ) )
5150impcom 431 . . . . . . . . . . . . . . . 16  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) )
5251imp 430 . . . . . . . . . . . . . . 15  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  { (
u `  0 ) ,  ( u ` 
1 ) }  e.  X )
5330, 38, 523jca 1185 . . . . . . . . . . . . . 14  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  ( ( # `
 u )  =  2  /\  ( u `
 0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )
54 eqcom 2438 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p  <->  p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 ) )
5539eqeq2d 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  <-> 
p  =  ( u `
 1 ) ) )
5655biimpd 210 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  ->  p  =  ( u `  1 ) ) )
5754, 56syl5bi 220 . . . . . . . . . . . . . . . . . . 19  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  1 )  =  p  ->  p  =  ( u ` 
1 ) ) )
5857com12 32 . . . . . . . . . . . . . . . . . 18  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p  ->  ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  ->  p  =  ( u `  1 ) ) )
5958ad2antll 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  =  u  ->  p  =  ( u `  1 ) ) )
6059com12 32 . . . . . . . . . . . . . . . 16  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  p  =  ( u `  1 ) ) )
6160adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  p  =  ( u `  1 ) ) )
6261imp 430 . . . . . . . . . . . . . 14  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  p  =  ( u `  1
) )
6322, 53, 62jca31 536 . . . . . . . . . . . . 13  |-  ( ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) ) )  /\  ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) ) )  ->  ( (
u  e. Word  V  /\  ( ( # `  u
)  =  2  /\  ( u `  0
)  =  P  /\  { ( u `  0
) ,  ( u `
 1 ) }  e.  X ) )  /\  p  =  ( u `  1 ) ) )
6463exp31 607 . . . . . . . . . . . 12  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X ) )  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( u  e. Word  V  /\  (
( # `  u )  =  2  /\  (
u `  0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )  /\  p  =  ( u `  1 ) ) ) ) )
6564eqcoms 2441 . . . . . . . . . . 11  |-  ( u  =  { <. 0 ,  P >. ,  <. 1 ,  p >. }  ->  (
( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( (
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( u  e. Word  V  /\  (
( # `  u )  =  2  /\  (
u `  0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )  /\  p  =  ( u `  1 ) ) ) ) )
6665impcom 431 . . . . . . . . . 10  |-  ( ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  /\  u  =  { <. 0 ,  P >. ,  <. 1 ,  p >. } )  ->  (
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  ( ( u  e. Word  V  /\  (
( # `  u )  =  2  /\  (
u `  0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )  /\  p  =  ( u `  1 ) ) ) )
6715, 66spcimedv 3171 . . . . . . . . 9  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( (
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  /\  ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2 )  /\  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p ) )  ->  E. u ( ( u  e. Word  V  /\  ( ( # `  u
)  =  2  /\  ( u `  0
)  =  P  /\  { ( u `  0
) ,  ( u `
 1 ) }  e.  X ) )  /\  p  =  ( u `  1 ) ) ) )
6813, 67mpd 15 . . . . . . . 8  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u
( ( u  e. Word  V  /\  ( ( # `  u )  =  2  /\  ( u ` 
0 )  =  P  /\  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) )  /\  p  =  ( u ` 
1 ) ) )
69 fveq2 5881 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  ( # `
 w )  =  ( # `  u
) )
7069eqeq1d 2431 . . . . . . . . . . . 12  |-  ( w  =  u  ->  (
( # `  w )  =  2  <->  ( # `  u
)  =  2 ) )
71 fveq1 5880 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  (
w `  0 )  =  ( u ` 
0 ) )
7271eqeq1d 2431 . . . . . . . . . . . 12  |-  ( w  =  u  ->  (
( w `  0
)  =  P  <->  ( u `  0 )  =  P ) )
73 fveq1 5880 . . . . . . . . . . . . . 14  |-  ( w  =  u  ->  (
w `  1 )  =  ( u ` 
1 ) )
7471, 73preq12d 4090 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( u ` 
0 ) ,  ( u `  1 ) } )
7574eleq1d 2498 . . . . . . . . . . . 12  |-  ( w  =  u  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )
7670, 72, 753anbi123d 1335 . . . . . . . . . . 11  |-  ( w  =  u  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 u )  =  2  /\  ( u `
 0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) ) )
7776elrab 3235 . . . . . . . . . 10  |-  ( u  e.  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  <->  ( u  e. Word  V  /\  ( (
# `  u )  =  2  /\  (
u `  0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) ) )
7877anbi1i 699 . . . . . . . . 9  |-  ( ( u  e.  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  /\  p  =  ( u `  1 ) )  <-> 
( ( u  e. Word  V  /\  ( ( # `  u )  =  2  /\  ( u ` 
0 )  =  P  /\  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) )  /\  p  =  ( u ` 
1 ) ) )
7978exbii 1714 . . . . . . . 8  |-  ( E. u ( u  e. 
{ w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }  /\  p  =  ( u ` 
1 ) )  <->  E. u
( ( u  e. Word  V  /\  ( ( # `  u )  =  2  /\  ( u ` 
0 )  =  P  /\  { ( u `
 0 ) ,  ( u `  1
) }  e.  X
) )  /\  p  =  ( u ` 
1 ) ) )
8068, 79sylibr 215 . . . . . . 7  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u
( u  e.  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) }  /\  p  =  ( u `  1 ) ) )
81 df-rex 2788 . . . . . . 7  |-  ( E. u  e.  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) } p  =  ( u ` 
1 )  <->  E. u
( u  e.  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) }  /\  p  =  ( u `  1 ) ) )
8280, 81sylibr 215 . . . . . 6  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u  e.  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } p  =  ( u `  1
) )
831rexeqi 3037 . . . . . 6  |-  ( E. u  e.  D  p  =  ( u ` 
1 )  <->  E. u  e.  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } p  =  ( u `  1
) )
8482, 83sylibr 215 . . . . 5  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u  e.  D  p  =  ( u `  1
) )
85 fveq1 5880 . . . . . . . 8  |-  ( t  =  u  ->  (
t `  1 )  =  ( u ` 
1 ) )
86 fvex 5891 . . . . . . . 8  |-  ( u `
 1 )  e. 
_V
8785, 3, 86fvmpt 5964 . . . . . . 7  |-  ( u  e.  D  ->  ( F `  u )  =  ( u ` 
1 ) )
8887eqeq2d 2443 . . . . . 6  |-  ( u  e.  D  ->  (
p  =  ( F `
 u )  <->  p  =  ( u `  1
) ) )
8988rexbiia 2933 . . . . 5  |-  ( E. u  e.  D  p  =  ( F `  u )  <->  E. u  e.  D  p  =  ( u `  1
) )
9084, 89sylibr 215 . . . 4  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u  e.  D  p  =  ( F `  u ) )
918, 90sylan2b 477 . . 3  |-  ( ( P  e.  V  /\  p  e.  R )  ->  E. u  e.  D  p  =  ( F `  u ) )
9291ralrimiva 2846 . 2  |-  ( P  e.  V  ->  A. p  e.  R  E. u  e.  D  p  =  ( F `  u ) )
93 dffo3 6052 . 2  |-  ( F : D -onto-> R  <->  ( F : D --> R  /\  A. p  e.  R  E. u  e.  D  p  =  ( F `  u ) ) )
945, 92, 93sylanbrc 668 1  |-  ( P  e.  V  ->  F : D -onto-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087   {cpr 4004   <.cop 4008    |-> cmpt 4484   -->wf 5597   -onto->wfo 5599   ` cfv 5601   0cc0 9538   1c1 9539   2c2 10659   #chash 12512  Word cword 12643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651
This theorem is referenced by:  wwlktovf1o  13013
  Copyright terms: Public domain W3C validator