MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  repswsymb Structured version   Unicode version

Theorem repswsymb 12816
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 12812 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
213adant3 1025 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  ( S repeatS  N )  =  ( x  e.  ( 0..^ N )  |->  S ) )
3 eqidd 2423 . 2  |-  ( ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  /\  x  =  I )  ->  S  =  S )
4 simp3 1007 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  I  e.  ( 0..^ N ) )
5 simp1 1005 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ( 0..^ N ) )  ->  S  e.  V )
62, 3, 4, 5fvmptd 5907 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872    |-> cmpt 4418   ` cfv 5537  (class class class)co 6242   0cc0 9483   NN0cn0 10813  ..^cfzo 11859   repeatS creps 12604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pr 4596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-reps 12612
This theorem is referenced by:  repswfsts  12823  repswlsw  12824  repswswrd  12826  repswccat  12827  repswrevw  12828  repsco  12875  repswpfx  38784
  Copyright terms: Public domain W3C validator