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.

Step	Hyp	Ref	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 ) )
3	2	bicomi 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 ) )
4	3	abbii 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 ) ) )
6	5	abbii 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 ) ) }
7	1, 4, 6	3eqtri 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
10	8, 9	eqeltri 2525	. . 3 |- { n | n e. ( 0..^ ( # ` W ) ) } e. _V
11		tru 1448	. . . . 5 |- T.
12	11, 11	pm3.2i 457	. . . 4 |- ( T. /\ T. )
13		ovex 6318	. . . . . . 7 |- ( W cyclShift n ) e. _V
14	13	a1i 11	. . . . . 6 |- ( T. -> ( W cyclShift n ) e. _V )
15		eqtr3 2472	. . . . . . . . . . . . 13 |- ( ( w = ( W cyclShift n ) /\ y = ( W cyclShift n ) ) -> w = y )
16	15	ex 436	. . . . . . . . . . . 12 |- ( w = ( W cyclShift n ) -> ( y = ( W cyclShift n ) -> w = y ) )
17	16	eqcoms 2459	. . . . . . . . . . 11 |- ( ( W cyclShift n ) = w -> ( y = ( W cyclShift n ) -> w = y ) )
18	17	adantl 468	. . . . . . . . . 10 |- ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> ( y = ( W cyclShift n ) -> w = y ) )
19	18	com12 32	. . . . . . . . 9 |- ( y = ( W cyclShift n ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
20	19	ad2antlr 733	. . . . . . . 8 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
21	20	alrimiv 1773	. . . . . . 7 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
22	21	ex 436	. . . . . 6 |- ( ( T. /\ y = ( W cyclShift n ) ) -> ( T. -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
23	14, 22	spcimedv 3133	. . . . 5 |- ( T. -> ( T. -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
24	23	imp 431	. . . 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 4523	. 2 |- { w | E. n ( n e. ( 0..^ ( # ` W ) ) /\ ( w e. Word V /\ ( W cyclShift n ) = w ) ) } e. _V
27	7, 26	eqeltri 2525	1 |- { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } e. _V

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)