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Theorem wrd2f1tovbij 12848
Description: There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
Assertion
Ref Expression
wrd2f1tovbij  |-  ( ( V  e.  Y  /\  P  e.  V )  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }
)
Distinct variable groups:    P, f, n, w    f, V, n, w    f, X, n, w
Allowed substitution hints:    Y( w, f, n)

Proof of Theorem wrd2f1tovbij
Dummy variables  p  t  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdexg 12510 . . . 4  |-  ( V  e.  Y  -> Word  V  e. 
_V )
21adantr 465 . . 3  |-  ( ( V  e.  Y  /\  P  e.  V )  -> Word  V  e.  _V )
3 rabexg 4590 . . 3  |-  (Word  V  e.  _V  ->  { t  e. Word  V  |  ( (
# `  t )  =  2  /\  (
t `  0 )  =  P  /\  { ( t `  0 ) ,  ( t ` 
1 ) }  e.  X ) }  e.  _V )
4 mptexg 6121 . . 3  |-  ( { t  e. Word  V  | 
( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) }  e.  _V  ->  (
x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) )  e.  _V )
52, 3, 43syl 20 . 2  |-  ( ( V  e.  Y  /\  P  e.  V )  ->  ( x  e.  {
t  e. Word  V  | 
( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) )  e.  _V )
6 fveq2 5857 . . . . . . 7  |-  ( w  =  u  ->  ( # `
 w )  =  ( # `  u
) )
76eqeq1d 2462 . . . . . 6  |-  ( w  =  u  ->  (
( # `  w )  =  2  <->  ( # `  u
)  =  2 ) )
8 fveq1 5856 . . . . . . 7  |-  ( w  =  u  ->  (
w `  0 )  =  ( u ` 
0 ) )
98eqeq1d 2462 . . . . . 6  |-  ( w  =  u  ->  (
( w `  0
)  =  P  <->  ( u `  0 )  =  P ) )
10 fveq1 5856 . . . . . . . 8  |-  ( w  =  u  ->  (
w `  1 )  =  ( u ` 
1 ) )
118, 10preq12d 4107 . . . . . . 7  |-  ( w  =  u  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( u ` 
0 ) ,  ( u `  1 ) } )
1211eleq1d 2529 . . . . . 6  |-  ( w  =  u  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
u `  0 ) ,  ( u ` 
1 ) }  e.  X ) )
137, 9, 123anbi123d 1294 . . . . 5  |-  ( w  =  u  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 u )  =  2  /\  ( u `
 0 )  =  P  /\  { ( u `  0 ) ,  ( u ` 
1 ) }  e.  X ) ) )
1413cbvrabv 3105 . . . 4  |-  { 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 ) }
15 preq2 4100 . . . . . 6  |-  ( n  =  p  ->  { P ,  n }  =  { P ,  p }
)
1615eleq1d 2529 . . . . 5  |-  ( n  =  p  ->  ( { P ,  n }  e.  X  <->  { P ,  p }  e.  X )
)
1716cbvrabv 3105 . . . 4  |-  { n  e.  V  |  { P ,  n }  e.  X }  =  {
p  e.  V  |  { P ,  p }  e.  X }
18 fveq2 5857 . . . . . . . 8  |-  ( t  =  w  ->  ( # `
 t )  =  ( # `  w
) )
1918eqeq1d 2462 . . . . . . 7  |-  ( t  =  w  ->  (
( # `  t )  =  2  <->  ( # `  w
)  =  2 ) )
20 fveq1 5856 . . . . . . . 8  |-  ( t  =  w  ->  (
t `  0 )  =  ( w ` 
0 ) )
2120eqeq1d 2462 . . . . . . 7  |-  ( t  =  w  ->  (
( t `  0
)  =  P  <->  ( w `  0 )  =  P ) )
22 fveq1 5856 . . . . . . . . 9  |-  ( t  =  w  ->  (
t `  1 )  =  ( w ` 
1 ) )
2320, 22preq12d 4107 . . . . . . . 8  |-  ( t  =  w  ->  { ( t `  0 ) ,  ( t ` 
1 ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
2423eleq1d 2529 . . . . . . 7  |-  ( t  =  w  ->  ( { ( t ` 
0 ) ,  ( t `  1 ) }  e.  X  <->  { (
w `  0 ) ,  ( w ` 
1 ) }  e.  X ) )
2519, 21, 243anbi123d 1294 . . . . . 6  |-  ( t  =  w  ->  (
( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X )  <->  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) ) )
2625cbvrabv 3105 . . . . 5  |-  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) }  =  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }
27 mpteq1 4520 . . . . 5  |-  ( { t  e. Word  V  | 
( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) }  =  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  ->  ( x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) )  =  ( x  e.  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  |->  ( x `  1 ) ) )
2826, 27ax-mp 5 . . . 4  |-  ( x  e.  { t  e. Word  V  |  ( ( # `
 t )  =  2  /\  ( t `
 0 )  =  P  /\  { ( t `  0 ) ,  ( t ` 
1 ) }  e.  X ) }  |->  ( x `  1 ) )  =  ( x  e.  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  |->  ( x `  1 ) )
2914, 17, 28wwlktovf1o 12847 . . 3  |-  ( P  e.  V  ->  (
x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) ) : {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X } )
3029adantl 466 . 2  |-  ( ( V  e.  Y  /\  P  e.  V )  ->  ( x  e.  {
t  e. Word  V  | 
( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) ) : {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X } )
31 f1oeq1 5798 . . 3  |-  ( f  =  ( x  e. 
{ t  e. Word  V  |  ( ( # `  t )  =  2  /\  ( t ` 
0 )  =  P  /\  { ( t `
 0 ) ,  ( t `  1
) }  e.  X
) }  |->  ( x `
 1 ) )  ->  ( f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }  <->  ( x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) ) : {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X } ) )
3231spcegv 3192 . 2  |-  ( ( x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  2  /\  ( t `  0
)  =  P  /\  { ( t `  0
) ,  ( t `
 1 ) }  e.  X ) } 
|->  ( x `  1
) )  e.  _V  ->  ( ( x  e. 
{ t  e. Word  V  |  ( ( # `  t )  =  2  /\  ( t ` 
0 )  =  P  /\  { ( t `
 0 ) ,  ( t `  1
) }  e.  X
) }  |->  ( x `
 1 ) ) : { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }
) )
335, 30, 32sylc 60 1  |-  ( ( V  e.  Y  /\  P  e.  V )  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   {crab 2811   _Vcvv 3106   {cpr 4022    |-> cmpt 4498   -1-1-onto->wf1o 5578   ` cfv 5579   0cc0 9481   1c1 9482   2c2 10574   #chash 12360  Word cword 12487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495
This theorem is referenced by:  rusgranumwwlkl1  24608
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