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Theorem wwlktovfo 30251
Description: Lemma 3 for wrd2f1tovbij 30253. (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 30249 . . 3  |-  F : D
--> R
54a1i 11 . 2  |-  ( P  e.  V  ->  F : D --> R )
62eleq2i 2506 . . . . 5  |-  ( p  e.  R  <->  p  e.  { n  e.  V  |  { P ,  n }  e.  X } )
7 preq2 3954 . . . . . . 7  |-  ( n  =  p  ->  { P ,  n }  =  { P ,  p }
)
87eleq1d 2508 . . . . . 6  |-  ( n  =  p  ->  ( { P ,  n }  e.  X  <->  { P ,  p }  e.  X )
)
98elrab 3116 . . . . 5  |-  ( p  e.  { n  e.  V  |  { P ,  n }  e.  X } 
<->  ( p  e.  V  /\  { P ,  p }  e.  X )
)
106, 9bitri 249 . . . 4  |-  ( p  e.  R  <->  ( p  e.  V  /\  { P ,  p }  e.  X
) )
11 simpl 457 . . . . . . . . . . 11  |-  ( ( p  e.  V  /\  { P ,  p }  e.  X )  ->  p  e.  V )
1211anim2i 569 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  ( P  e.  V  /\  p  e.  V ) )
13 eqidd 2443 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  { <. 0 ,  P >. , 
<. 1 ,  p >. } )
14 wrdlen2i 12545 . . . . . . . . . 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 ) ) ) )
1512, 13, 14sylc 60 . . . . . . . . 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 ) ) )
16 prex 4533 . . . . . . . . . . 11  |-  { <. 0 ,  P >. , 
<. 1 ,  p >. }  e.  _V
1716a1i 11 . . . . . . . . . 10  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  { <. 0 ,  P >. ,  <. 1 ,  p >. }  e.  _V )
18 eleq1 2502 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  <->  u  e. Word  V ) )
1918biimpd 207 . . . . . . . . . . . . . . . . . . 19  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. }  e. Word  V  ->  u  e. Word  V ) )
2019adantr 465 . . . . . . . . . . . . . . . . . 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 )
)
2120com12 31 . . . . . . . . . . . . . . . . 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 ) )
2221adantr 465 . . . . . . . . . . . . . . . 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 ) )
2322adantr 465 . . . . . . . . . . . . . . 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 ) )
2423impcom 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 ) ) )  ->  u  e. Word  V )
25 fveq2 5690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  (
# `  u )
)
2625eqeq1d 2450 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  <-> 
( # `  u )  =  2 ) )
2726biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  ->  ( # `  u
)  =  2 ) )
2827adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  (
( # `  { <. 0 ,  P >. , 
<. 1 ,  p >. } )  =  2  ->  ( # `  u
)  =  2 ) )
2928com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  { <. 0 ,  P >. ,  <. 1 ,  p >. } )  =  2  ->  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  /\  ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
) )  ->  ( # `
 u )  =  2 ) )
3029adantl 466 . . . . . . . . . . . . . . . . 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 ) )
3130adantr 465 . . . . . . . . . . . . . . . 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 ) )
3231impcom 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
)  =  2 )
33 fveq1 5689 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  0
)  =  ( u `
 0 ) )
3433eqeq1d 2450 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  0 )  =  P  <->  ( u `  0 )  =  P ) )
3534biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  0 )  =  P  ->  (
u `  0 )  =  P ) )
3635adantr 465 . . . . . . . . . . . . . . . . . . 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 ) )
3736com12 31 . . . . . . . . . . . . . . . . . 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 ) )
3837adantr 465 . . . . . . . . . . . . . . . . 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 ) )
3938adantl 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 `  0 )  =  P ) )
4039impcom 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 )  =  P )
41 fveq1 5689 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1
)  =  ( u `
 1 ) )
4241eqeq1d 2450 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  1 )  =  p  <->  ( u `  1 )  =  p ) )
4334, 42anbi12d 710 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( ( {
<. 0 ,  P >. ,  <. 1 ,  p >. } `  0 )  =  P  /\  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p )  <->  ( (
u `  0 )  =  P  /\  (
u `  1 )  =  p ) ) )
44 preq12 3955 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  { ( u `
 0 ) ,  ( u `  1
) }  =  { P ,  p }
)
4544eqcomd 2447 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  { P ,  p }  =  {
( u `  0
) ,  ( u `
 1 ) } )
4645eleq1d 2508 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  ( { P ,  p }  e.  X  <->  { ( u `  0
) ,  ( u `
 1 ) }  e.  X ) )
4746biimpd 207 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( u `  0
)  =  P  /\  ( u `  1
)  =  p )  ->  ( { P ,  p }  e.  X  ->  { ( u ` 
0 ) ,  ( u `  1 ) }  e.  X ) )
4843, 47syl6bi 228 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
4948com12 31 . . . . . . . . . . . . . . . . . . . 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 ) ) )
5049adantl 466 . . . . . . . . . . . . . . . . . . 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 ) ) )
5150com13 80 . . . . . . . . . . . . . . . . . 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
) ) )
5251ad2antll 728 . . . . . . . . . . . . . . . . 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
) ) )
5352impcom 430 . . . . . . . . . . . . . . . 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
) )
5453imp 429 . . . . . . . . . . . . . . 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 )
5532, 40, 543jca 1168 . . . . . . . . . . . . . 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 ) )
56 eqcom 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p  <->  p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 ) )
5741eqeq2d 2453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  <-> 
p  =  ( u `
 1 ) ) )
5857biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( p  =  ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  ->  p  =  ( u `  1 ) ) )
5956, 58syl5bi 217 . . . . . . . . . . . . . . . . . . 19  |-  ( {
<. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  -> 
( ( { <. 0 ,  P >. , 
<. 1 ,  p >. } `  1 )  =  p  ->  p  =  ( u ` 
1 ) ) )
6059com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( ( { <. 0 ,  P >. ,  <. 1 ,  p >. } `  1 )  =  p  ->  ( { <. 0 ,  P >. ,  <. 1 ,  p >. }  =  u  ->  p  =  ( u `  1 ) ) )
6160ad2antll 728 . . . . . . . . . . . . . . . . 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 ) ) )
6261com12 31 . . . . . . . . . . . . . . . 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 ) ) )
6362adantr 465 . . . . . . . . . . . . . . 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 ) ) )
6463imp 429 . . . . . . . . . . . . . 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
) )
6524, 55, 64jca31 534 . . . . . . . . . . . . 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 ) ) )
6665exp31 604 . . . . . . . . . . . 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 ) ) ) ) )
6766eqcoms 2445 . . . . . . . . . . 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 ) ) ) ) )
6867impcom 430 . . . . . . . . . 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 ) ) ) )
6917, 68spcimedv 3055 . . . . . . . . 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 ) ) ) )
7015, 69mpd 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 ) ) )
71 fveq2 5690 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  ( # `
 w )  =  ( # `  u
) )
7271eqeq1d 2450 . . . . . . . . . . . 12  |-  ( w  =  u  ->  (
( # `  w )  =  2  <->  ( # `  u
)  =  2 ) )
73 fveq1 5689 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  (
w `  0 )  =  ( u ` 
0 ) )
7473eqeq1d 2450 . . . . . . . . . . . 12  |-  ( w  =  u  ->  (
( w `  0
)  =  P  <->  ( u `  0 )  =  P ) )
75 fveq1 5689 . . . . . . . . . . . . . 14  |-  ( w  =  u  ->  (
w `  1 )  =  ( u ` 
1 ) )
7673, 75preq12d 3961 . . . . . . . . . . . . 13  |-  ( w  =  u  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( u ` 
0 ) ,  ( u `  1 ) } )
7776eleq1d 2508 . . . . . . . . . . . 12  |-  ( w  =  u  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )
7872, 74, 773anbi123d 1289 . . . . . . . . . . 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 ) ) )
7978elrab 3116 . . . . . . . . . 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 ) ) )
8079anbi1i 695 . . . . . . . . 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 ) ) )
8180exbii 1634 . . . . . . . 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 ) ) )
8270, 81sylibr 212 . . . . . . 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 ) ) )
83 df-rex 2720 . . . . . . 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 ) ) )
8482, 83sylibr 212 . . . . . 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
) )
851rexeqi 2921 . . . . . 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
) )
8684, 85sylibr 212 . . . . 5  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u  e.  D  p  =  ( u `  1
) )
87 fvex 5700 . . . . . . . 8  |-  ( u `
 1 )  e. 
_V
88 fveq1 5689 . . . . . . . . 9  |-  ( t  =  u  ->  (
t `  1 )  =  ( u ` 
1 ) )
8988, 3fvmptg 5771 . . . . . . . 8  |-  ( ( u  e.  D  /\  ( u `  1
)  e.  _V )  ->  ( F `  u
)  =  ( u `
 1 ) )
9087, 89mpan2 671 . . . . . . 7  |-  ( u  e.  D  ->  ( F `  u )  =  ( u ` 
1 ) )
9190eqeq2d 2453 . . . . . 6  |-  ( u  e.  D  ->  (
p  =  ( F `
 u )  <->  p  =  ( u `  1
) ) )
9291rexbiia 2747 . . . . 5  |-  ( E. u  e.  D  p  =  ( F `  u )  <->  E. u  e.  D  p  =  ( u `  1
) )
9386, 92sylibr 212 . . . 4  |-  ( ( P  e.  V  /\  ( p  e.  V  /\  { P ,  p }  e.  X )
)  ->  E. u  e.  D  p  =  ( F `  u ) )
9410, 93sylan2b 475 . . 3  |-  ( ( P  e.  V  /\  p  e.  R )  ->  E. u  e.  D  p  =  ( F `  u ) )
9594ralrimiva 2798 . 2  |-  ( P  e.  V  ->  A. p  e.  R  E. u  e.  D  p  =  ( F `  u ) )
96 dffo3 5857 . 2  |-  ( F : D -onto-> R  <->  ( F : D --> R  /\  A. p  e.  R  E. u  e.  D  p  =  ( F `  u ) ) )
975, 95, 96sylanbrc 664 1  |-  ( P  e.  V  ->  F : D -onto-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 2971   {cpr 3878   <.cop 3882    e. cmpt 4349   -->wf 5413   -onto->wfo 5415   ` cfv 5417   0cc0 9281   1c1 9282   2c2 10370   #chash 12102  Word cword 12220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228
This theorem is referenced by:  wwlktovf1o  30252
  Copyright terms: Public domain W3C validator