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Theorem repswsymb 12707
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 12703 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
213adant3 1015 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  ( S repeatS  N )  =  ( x  e.  ( 0..^ N )  |->  S ) )
3 eqidd 2401 . 2  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  /\  x  =  I )  ->  S  =  S )
4 simp3 997 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  I  e.  ( 0..^ N ) )
5 simp1 995 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  S  e.  V )
62, 3, 4, 5fvmptd 5892 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   0cc0 9440   NN0cn0 10754  ..^cfzo 11765   repeatS creps 12495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-reps 12503
This theorem is referenced by:  repswfsts  12714  repswlsw  12715  repswswrd  12717  repswccat  12718  repswrevw  12719  repsco  12766  repswpfx  37855
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