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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 Word ..^ cyclShift
Distinct variable groups:   ,   ,,
Allowed substitution hint:   ()

Proof of Theorem cshwsexa
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-rab 2746 . . 3 Word ..^ cyclShift Word ..^ cyclShift
2 r19.42v 2945 . . . . 5 ..^ Word cyclShift Word ..^ cyclShift
32bicomi 206 . . . 4 Word ..^ cyclShift ..^ Word cyclShift
43abbii 2567 . . 3 Word ..^ cyclShift ..^ Word cyclShift
5 df-rex 2743 . . . 4 ..^ Word cyclShift ..^ Word cyclShift
65abbii 2567 . . 3 ..^ Word cyclShift ..^ Word cyclShift
71, 4, 63eqtri 2477 . 2 Word ..^ cyclShift ..^ Word cyclShift
8 abid2 2573 . . . 4 ..^ ..^
9 ovex 6318 . . . 4 ..^
108, 9eqeltri 2525 . . 3 ..^
11 tru 1448 . . . . 5
1211, 11pm3.2i 457 . . . 4
13 ovex 6318 . . . . . . 7 cyclShift
1413a1i 11 . . . . . 6 cyclShift
15 eqtr3 2472 . . . . . . . . . . . . 13 cyclShift cyclShift
1615ex 436 . . . . . . . . . . . 12 cyclShift cyclShift
1716eqcoms 2459 . . . . . . . . . . 11 cyclShift cyclShift
1817adantl 468 . . . . . . . . . 10 Word cyclShift cyclShift
1918com12 32 . . . . . . . . 9 cyclShift Word cyclShift
2019ad2antlr 733 . . . . . . . 8 cyclShift Word cyclShift
2120alrimiv 1773 . . . . . . 7 cyclShift Word cyclShift
2221ex 436 . . . . . 6 cyclShift Word cyclShift
2314, 22spcimedv 3133 . . . . 5 Word cyclShift
2423imp 431 . . . 4 Word cyclShift
2512, 24mp1i 13 . . 3 ..^ Word cyclShift
2610, 25zfrep4 4523 . 2 ..^ Word cyclShift
277, 26eqeltri 2525 1 Word ..^ cyclShift
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371  wal 1442   wceq 1444   wtru 1445  wex 1663   wcel 1887  cab 2437  wrex 2738  crab 2741  cvv 3045  cfv 5582  (class class class)co 6290  cc0 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)
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