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Theorem cshwsexa 12562
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 2804 . . 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 2973 . . . . 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 202 . . . 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 2585 . . 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 2801 . . . 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 2585 . . 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 2484 . 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 2591 . . . 4  |-  { n  |  n  e.  (
0..^ ( # `  W
) ) }  =  ( 0..^ ( # `  W
) )
9 ovex 6217 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
108, 9eqeltri 2535 . . 3  |-  { n  |  n  e.  (
0..^ ( # `  W
) ) }  e.  _V
11 tru 1374 . . . . 5  |- T.
1211, 11pm3.2i 455 . . . 4  |-  ( T. 
/\ T.  )
13 ovex 6217 . . . . . . 7  |-  ( W cyclShift  n )  e.  _V
1413a1i 11 . . . . . 6  |-  ( T. 
->  ( W cyclShift  n )  e.  _V )
15 eqtr3 2479 . . . . . . . . . . . . 13  |-  ( ( w  =  ( W cyclShift  n )  /\  y  =  ( W cyclShift  n ) )  ->  w  =  y )
1615ex 434 . . . . . . . . . . . 12  |-  ( w  =  ( W cyclShift  n )  ->  ( y  =  ( W cyclShift  n )  ->  w  =  y ) )
1716eqcoms 2463 . . . . . . . . . . 11  |-  ( ( W cyclShift  n )  =  w  ->  ( y  =  ( W cyclShift  n )  ->  w  =  y ) )
1817adantl 466 . . . . . . . . . 10  |-  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  (
y  =  ( W cyclShift  n )  ->  w  =  y ) )
1918com12 31 . . . . . . . . 9  |-  ( y  =  ( W cyclShift  n )  ->  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2019ad2antlr 726 . . . . . . . 8  |-  ( ( ( T.  /\  y  =  ( W cyclShift  n ) )  /\ T.  )  ->  ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2120alrimiv 1686 . . . . . . 7  |-  ( ( ( T.  /\  y  =  ( W cyclShift  n ) )  /\ T.  )  ->  A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2221ex 434 . . . . . 6  |-  ( ( T.  /\  y  =  ( W cyclShift  n )
)  ->  ( T.  ->  A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) ) )
2314, 22spcimedv 3154 . . . . 5  |-  ( T. 
->  ( T.  ->  E. y A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) ) )
2423imp 429 . . . 4  |-  ( ( T.  /\ T.  )  ->  E. y A. w
( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y )
)
2512, 24mp1i 12 . . 3  |-  ( n  e.  ( 0..^ (
# `  W )
)  ->  E. y A. w ( ( w  e. Word  V  /\  ( W cyclShift  n )  =  w )  ->  w  =  y ) )
2610, 25zfrep4 4511 . 2  |-  { w  |  E. n ( n  e.  ( 0..^ (
# `  W )
)  /\  ( w  e. Word  V  /\  ( W cyclShift  n )  =  w ) ) }  e.  _V
277, 26eqeltri 2535 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 369   A.wal 1368    = wceq 1370   T. wtru 1371   E.wex 1587    e. wcel 1758   {cab 2436   E.wrex 2796   {crab 2799   _Vcvv 3070   ` cfv 5518  (class class class)co 6192   0cc0 9385  ..^cfzo 11651   #chash 12206  Word cword 12325   cyclShift ccsh 12529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-nul 4521
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980  df-uni 4192  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by: (None)
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