Theorem cshwsexa 13570

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 2921	. . 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 3092	. . . . 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 214	. . . 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 2739	. . 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 2918	. . . 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 2739	. . 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 2648	. 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 2745	. . . 4 |- { n | n e. ( 0..^ ( # ` W ) ) } = ( 0..^ ( # ` W ) )
9		ovex 6678	. . . 4 |- ( 0..^ ( # ` W ) ) e. _V
10	8, 9	eqeltri 2697	. . 3 |- { n | n e. ( 0..^ ( # ` W ) ) } e. _V
11		tru 1487	. . . . 5 |- T.
12	11, 11	pm3.2i 471	. . . 4 |- ( T. /\ T. )
13		ovexd 6680	. . . . . 6 |- ( T. -> ( W cyclShift n ) e. _V )
14		eqtr3 2643	. . . . . . . . . . . . 13 |- ( ( w = ( W cyclShift n ) /\ y = ( W cyclShift n ) ) -> w = y )
15	14	ex 450	. . . . . . . . . . . 12 |- ( w = ( W cyclShift n ) -> ( y = ( W cyclShift n ) -> w = y ) )
16	15	eqcoms 2630	. . . . . . . . . . 11 |- ( ( W cyclShift n ) = w -> ( y = ( W cyclShift n ) -> w = y ) )
17	16	adantl 482	. . . . . . . . . 10 |- ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> ( y = ( W cyclShift n ) -> w = y ) )
18	17	com12 32	. . . . . . . . 9 |- ( y = ( W cyclShift n ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
19	18	ad2antlr 763	. . . . . . . 8 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
20	19	alrimiv 1855	. . . . . . 7 |- ( ( ( T. /\ y = ( W cyclShift n ) ) /\ T. ) -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
21	20	ex 450	. . . . . 6 |- ( ( T. /\ y = ( W cyclShift n ) ) -> ( T. -> A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
22	13, 21	spcimedv 3292	. . . . 5 |- ( T. -> ( T. -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) ) )
23	22	imp 445	. . . 4 |- ( ( T. /\ T. ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
24	12, 23	mp1i 13	. . 3 |- ( n e. ( 0..^ ( # ` W ) ) -> E. y A. w ( ( w e. Word V /\ ( W cyclShift n ) = w ) -> w = y ) )
25	10, 24	zfrep4 4779	. 2 |- { w | E. n ( n e. ( 0..^ ( # ` W ) ) /\ ( w e. Word V /\ ( W cyclShift n ) = w ) ) } e. _V
26	7, 25	eqeltri 2697	1 |- { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } e. _V

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)