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Theorem repswsymb 12533
Description: The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
repswsymb  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  I
)  =  S )

Proof of Theorem repswsymb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reps 12529 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
213adant3 1008 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  ( S repeatS  N )  =  ( x  e.  ( 0..^ N )  |->  S ) )
3 eqidd 2455 . 2  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  /\  x  =  I )  ->  S  =  S )
4 simp3 990 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  I  e.  ( 0..^ N ) )
5 simp1 988 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  S  e.  V )
62, 3, 4, 5fvmptd 5891 1  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  ( ( S repeatS  N ) `  I
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   0cc0 9396   NN0cn0 10693  ..^cfzo 11668   repeatS creps 12349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-reps 12357
This theorem is referenced by:  repswfsts  12540  repswlsw  12541  repswswrd  12543  repswccat  12544  repswrevw  12545  repsco  12588
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