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Theorem cshwsexa 12923
Description: The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.)
Assertion
Ref Expression
cshwsexa  |-  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  e.  _V
Distinct variable groups:    n, V    n, W, w
Allowed substitution hint:    V( w)

Proof of Theorem cshwsexa
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-rab 2746 . . 3  |-  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  =  { w  |  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) }
2 r19.42v 2945 . . . . 5  |-  ( E. n  e.  ( 0..^ ( # `  W
) ) ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  <->  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) )
32bicomi 206 . . . 4  |-  ( ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W
) ) ( W cyclShift  n )  =  w )  <->  E. n  e.  ( 0..^ ( # `  W
) ) ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) )
43abbii 2567 . . 3  |-  { w  |  ( w  e. Word  V  /\  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w ) }  =  {
w  |  E. n  e.  ( 0..^ ( # `  W ) ) ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) }
5 df-rex 2743 . . . 4  |-  ( E. n  e.  ( 0..^ ( # `  W
) ) ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  <->  E. n ( n  e.  ( 0..^ (
# `  W )
)  /\  ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) ) )
65abbii 2567 . . 3  |-  { w  |  E. n  e.  ( 0..^ ( # `  W
) ) ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) }  =  {
w  |  E. n
( n  e.  ( 0..^ ( # `  W
) )  /\  (
w  e. Word  V  /\  ( W cyclShift  n )  =  w ) ) }
71, 4, 63eqtri 2477 . 2  |-  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  =  { w  |  E. n ( n  e.  ( 0..^ (
# `  W )
)  /\  ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) ) }
8 abid2 2573 . . . 4  |-  { n  |  n  e.  (
0..^ ( # `  W
) ) }  =  ( 0..^ ( # `  W
) )
9 ovex 6318 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
108, 9eqeltri 2525 . . 3  |-  { n  |  n  e.  (
0..^ ( # `  W
) ) }  e.  _V
11 tru 1448 . . . . 5  |- T.
1211, 11pm3.2i 457 . . . 4  |-  ( T. 
/\ T.  )
13 ovex 6318 . . . . . . 7  |-  ( W cyclShift  n )  e.  _V
1413a1i 11 . . . . . 6  |-  ( T. 
->  ( W cyclShift  n )  e.  _V )
15 eqtr3 2472 . . . . . . . . . . . . 13  |-  ( ( w  =  ( W cyclShift  n )  /\  y  =  ( W cyclShift  n ) )  ->  w  =  y )
1615ex 436 . . . . . . . . . . . 12  |-  ( w  =  ( W cyclShift  n )  ->  ( y  =  ( W cyclShift  n )  ->  w  =  y ) )
1716eqcoms 2459 . . . . . . . . . . 11  |-  ( ( W cyclShift  n )  =  w  ->  ( y  =  ( W cyclShift  n )  ->  w  =  y ) )
1817adantl 468 . . . . . . . . . 10  |-  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  (
y  =  ( W cyclShift  n )  ->  w  =  y ) )
1918com12 32 . . . . . . . . 9  |-  ( y  =  ( W cyclShift  n )  ->  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2019ad2antlr 733 . . . . . . . 8  |-  ( ( ( T.  /\  y  =  ( W cyclShift  n ) )  /\ T.  )  ->  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2120alrimiv 1773 . . . . . . 7  |-  ( ( ( T.  /\  y  =  ( W cyclShift  n ) )  /\ T.  )  ->  A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2221ex 436 . . . . . 6  |-  ( ( T.  /\  y  =  ( W cyclShift  n )
)  ->  ( T.  ->  A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) ) )
2314, 22spcimedv 3133 . . . . 5  |-  ( T. 
->  ( T.  ->  E. y A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) ) )
2423imp 431 . . . 4  |-  ( ( T.  /\ T.  )  ->  E. y A. w
( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y )
)
2512, 24mp1i 13 . . 3  |-  ( n  e.  ( 0..^ (
# `  W )
)  ->  E. y A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2610, 25zfrep4 4523 . 2  |-  { w  |  E. n ( n  e.  ( 0..^ (
# `  W )
)  /\  ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) ) }  e.  _V
277, 26eqeltri 2525 1  |-  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442    = wceq 1444   T. wtru 1445   E.wex 1663    e. wcel 1887   {cab 2437   E.wrex 2738   {crab 2741   _Vcvv 3045   ` cfv 5582  (class class class)co 6290   0cc0 9539  ..^cfzo 11915   #chash 12515  Word cword 12656   cyclShift ccsh 12890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-nul 4534
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-pr 3971  df-uni 4199  df-iota 5546  df-fv 5590  df-ov 6293
This theorem is referenced by: (None)
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