Theorem cshwsexa 12755

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.

Step	Hyp	Ref	Expression
1		df-rab 2762	. . 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 2961	. . . . 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 ) )
3	2	bicomi 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 ) )
4	3	abbii 2536	. . 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 2759	. . . 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 ) ) )
6	5	abbii 2536	. . 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 ) ) }
7	1, 4, 6	3eqtri 2435	. 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 2542	. . . 4 |- { n | n e. ( 0..^ ( # ` W ) ) } = ( 0..^ ( # ` W ) )
9		ovex 6262	. . . 4 |- ( 0..^ ( # ` W ) ) e. _V
10	8, 9	eqeltri 2486	. . 3 |- { n | n e. ( 0..^ ( # ` W ) ) } e. _V
11		tru 1409	. . . . 5 |- T.
12	11, 11	pm3.2i 453	. . . 4 |- ( T. /\ T. )
13		ovex 6262	. . . . . . 7 |- ( W cyclShift n ) e. _V
14	13	a1i 11	. . . . . 6 |- ( T. -> ( W cyclShift n ) e. _V )
15		eqtr3 2430	. . . . . . . . . . . . 13 |- ( ( w = ( W cyclShift n ) /\ y = ( W cyclShift n ) ) -> w = y )
16	15	ex 432	. . . . . . . . . . . 12 |- ( w = ( W cyclShift n ) -> ( y = ( W cyclShift n ) -> w = y ) )
17	16	eqcoms 2414	. . . . . . . . . . 11 |- ( ( W cyclShift n ) = w -> ( y = ( W cyclShift n ) -> w = y ) )
18	17	adantl 464	. . . . . . . . . 10 |- ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> ( y = ( W cyclShift n ) -> w = y ) )
19	18	com12 29	. . . . . . . . 9 |- ( y = ( W cyclShift n ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
20	19	ad2antlr 725	. . . . . . . 8 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
21	20	alrimiv 1740	. . . . . . 7 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
22	21	ex 432	. . . . . 6 |- ( ( T. /\ y = ( W cyclShift n ) ) -> ( T. -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
23	14, 22	spcimedv 3142	. . . . 5 |- ( T. -> ( T. -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
24	23	imp 427	. . . 4 |- ( ( T. /\ T. ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
25	12, 24	mp1i 13	. . 3 |- ( n e. ( 0..^ ( # ` W ) ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
26	10, 25	zfrep4 4514	. 2 |- { w | E. n ( n e. ( 0..^ ( # ` W ) ) /\ ( w e. Word V /\ ( W cyclShift n ) = w ) ) } e. _V
27	7, 26	eqeltri 2486	1 |- { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } e. _V

Syntax hints:   -> wi 4   /\ wa 367   A.wal 1403   = wceq 1405   T. wtru 1406   E.wex 1633   e. wcel 1842   {cab 2387   E.wrex 2754   {crab 2757   _Vcvv 3058   ` cfv 5525  (class class class)co 6234   0cc0 9442  ..^cfzo 11767   #chash 12359  Word cword 12490   cyclShift ccsh 12722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-nul 4524

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-sn 3972  df-pr 3974  df-uni 4191  df-iota 5489  df-fv 5533  df-ov 6237

This theorem is referenced by: (None)