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Theorem iswrd 12554
Description: Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)
Assertion
Ref Expression
iswrd  |-  ( W  e. Word  S  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S )
Distinct variable groups:    S, l    W, l

Proof of Theorem iswrd
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-word 12546 . . 3  |- Word  S  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }
21eleq2i 2535 . 2  |-  ( W  e. Word  S  <->  W  e.  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S } )
3 ovex 6324 . . . . 5  |-  ( 0..^ l )  e.  _V
4 fex 6146 . . . . 5  |-  ( ( W : ( 0..^ l ) --> S  /\  ( 0..^ l )  e. 
_V )  ->  W  e.  _V )
53, 4mpan2 671 . . . 4  |-  ( W : ( 0..^ l ) --> S  ->  W  e.  _V )
65rexlimivw 2946 . . 3  |-  ( E. l  e.  NN0  W : ( 0..^ l ) --> S  ->  W  e.  _V )
7 feq1 5719 . . . 4  |-  ( w  =  W  ->  (
w : ( 0..^ l ) --> S  <->  W :
( 0..^ l ) --> S ) )
87rexbidv 2968 . . 3  |-  ( w  =  W  ->  ( E. l  e.  NN0  w : ( 0..^ l ) --> S  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S ) )
96, 8elab3 3253 . 2  |-  ( W  e.  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S )
102, 9bitri 249 1  |-  ( W  e. Word  S  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   -->wf 5590  (class class class)co 6296   0cc0 9509   NN0cn0 10816  ..^cfzo 11821  Word cword 12538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-word 12546
This theorem is referenced by:  iswrdi  12557  wrdf  12558  sswrdOLD  12562  cshword  12774  motcgrg  24057  cshword2  32555
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