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Theorem repsf 12739
Description: The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
Assertion
Ref Expression
repsf  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N ) : ( 0..^ N ) --> V )

Proof of Theorem repsf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 455 . . . . 5  |-  ( ( S  e.  V  /\  x  e.  ( 0..^ N ) )  ->  S  e.  V )
21ralrimiva 2868 . . . 4  |-  ( S  e.  V  ->  A. x  e.  ( 0..^ N ) S  e.  V )
32adantr 463 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  ->  A. x  e.  (
0..^ N ) S  e.  V )
4 eqid 2454 . . . 4  |-  ( x  e.  ( 0..^ N )  |->  S )  =  ( x  e.  ( 0..^ N )  |->  S )
54fmpt 6028 . . 3  |-  ( A. x  e.  ( 0..^ N ) S  e.  V  <->  ( x  e.  ( 0..^ N ) 
|->  S ) : ( 0..^ N ) --> V )
63, 5sylib 196 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( x  e.  ( 0..^ N )  |->  S ) : ( 0..^ N ) --> V )
7 reps 12736 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  =  ( x  e.  ( 0..^ N ) 
|->  S ) )
87feq1d 5699 . 2  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( ( S repeatS  N
) : ( 0..^ N ) --> V  <->  ( x  e.  ( 0..^ N ) 
|->  S ) : ( 0..^ N ) --> V ) )
96, 8mpbird 232 1  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N ) : ( 0..^ N ) --> V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804    |-> cmpt 4497   -->wf 5566  (class class class)co 6270   0cc0 9481   NN0cn0 10791  ..^cfzo 11799   repeatS creps 12528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-reps 12536
This theorem is referenced by:  repsw  12741  repswlen  12742  repswswrd  12750  repsco  12799
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