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Theorem isword 25983
Description: The words over a set  A. (Contributed by FL, 14-Jan-2014.)
Assertion
Ref Expression
isword  |-  ( ( A  e.  B  /\  N  e.  NN0 )  -> 
( A  Words  N )  =  ( A  ^m  ( 1 ... N
) ) )

Proof of Theorem isword
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
21adantr 451 . 2  |-  ( ( A  e.  B  /\  N  e.  NN0 )  ->  A  e.  _V )
3 simpr 447 . 2  |-  ( ( A  e.  B  /\  N  e.  NN0 )  ->  N  e.  NN0 )
4 ovex 5883 . . 3  |-  ( A  ^m  ( 1 ... N ) )  e. 
_V
54a1i 10 . 2  |-  ( ( A  e.  B  /\  N  e.  NN0 )  -> 
( A  ^m  (
1 ... N ) )  e.  _V )
6 oveq1 5865 . . 3  |-  ( x  =  A  ->  (
x  ^m  ( 1 ... y ) )  =  ( A  ^m  ( 1 ... y
) ) )
7 oveq2 5866 . . . 4  |-  ( y  =  N  ->  (
1 ... y )  =  ( 1 ... N
) )
87oveq2d 5874 . . 3  |-  ( y  =  N  ->  ( A  ^m  ( 1 ... y ) )  =  ( A  ^m  (
1 ... N ) ) )
9 df-words 25982 . . 3  |-  Words  =  ( x  e.  _V , 
y  e.  NN0  |->  ( x  ^m  ( 1 ... y ) ) )
106, 8, 9ovmpt2g 5982 . 2  |-  ( ( A  e.  _V  /\  N  e.  NN0  /\  ( A  ^m  ( 1 ... N ) )  e. 
_V )  ->  ( A  Words  N )  =  ( A  ^m  (
1 ... N ) ) )
112, 3, 5, 10syl3anc 1182 1  |-  ( ( A  e.  B  /\  N  e.  NN0 )  -> 
( A  Words  N )  =  ( A  ^m  ( 1 ... N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858    ^m cmap 6772   1c1 8738   NN0cn0 9965   ...cfz 10782    Words cwrd 25981
This theorem is referenced by:  isnword  25986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-words 25982
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