Theorem cshwsexa 12751

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 2823	. . 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 3016	. . . . 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 2601	. . 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 2820	. . . 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 2601	. . 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 2500	. 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 2607	. . . 4 |- { n | n e. ( 0..^ ( # ` W ) ) } = ( 0..^ ( # ` W ) )
9		ovex 6307	. . . 4 |- ( 0..^ ( # ` W ) ) e. _V
10	8, 9	eqeltri 2551	. . 3 |- { n | n e. ( 0..^ ( # ` W ) ) } e. _V
11		tru 1383	. . . . 5 |- T.
12	11, 11	pm3.2i 455	. . . 4 |- ( T. /\ T. )
13		ovex 6307	. . . . . . 7 |- ( W cyclShift n ) e. _V
14	13	a1i 11	. . . . . 6 |- ( T. -> ( W cyclShift n ) e. _V )
15		eqtr3 2495	. . . . . . . . . . . . 13 |- ( ( w = ( W cyclShift n ) /\ y = ( W cyclShift n ) ) -> w = y )
16	15	ex 434	. . . . . . . . . . . 12 |- ( w = ( W cyclShift n ) -> ( y = ( W cyclShift n ) -> w = y ) )
17	16	eqcoms 2479	. . . . . . . . . . 11 |- ( ( W cyclShift n ) = w -> ( y = ( W cyclShift n ) -> w = y ) )
18	17	adantl 466	. . . . . . . . . 10 |- ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> ( y = ( W cyclShift n ) -> w = y ) )
19	18	com12 31	. . . . . . . . 9 |- ( y = ( W cyclShift n ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
20	19	ad2antlr 726	. . . . . . . 8 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
21	20	alrimiv 1695	. . . . . . 7 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
22	21	ex 434	. . . . . 6 |- ( ( T. /\ y = ( W cyclShift n ) ) -> ( T. -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
23	14, 22	spcimedv 3197	. . . . 5 |- ( T. -> ( T. -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
24	23	imp 429	. . . 4 |- ( ( T. /\ T. ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
25	12, 24	mp1i 12	. . 3 |- ( n e. ( 0..^ ( # ` W ) ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
26	10, 25	zfrep4 4566	. 2 |- { w | E. n ( n e. ( 0..^ ( # ` W ) ) /\ ( w e. Word V /\ ( W cyclShift n ) = w ) ) } e. _V
27	7, 26	eqeltri 2551	1 |- { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } e. _V

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1377   = wceq 1379   T. wtru 1380   E.wex 1596   e. wcel 1767   {cab 2452   E.wrex 2815   {crab 2818   _Vcvv 3113   ` cfv 5586  (class class class)co 6282   0cc0 9488  ..^cfzo 11788   #chash 12369  Word cword 12496   cyclShift ccsh 12718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-nul 4576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5549  df-fv 5594  df-ov 6285

This theorem is referenced by: (None)