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Theorem repsundef 12722
Description: A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.)
Assertion
Ref Expression
repsundef  |-  ( N  e/  NN0  ->  ( S repeatS  N )  =  (/) )

Proof of Theorem repsundef
Dummy variables  n  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-reps 12529 . . 3  |- repeatS  =  ( s  e.  _V ,  n  e.  NN0  |->  ( x  e.  ( 0..^ n )  |->  s ) )
2 ovex 6320 . . . 4  |-  ( 0..^ n )  e.  _V
32mptex 6142 . . 3  |-  ( x  e.  ( 0..^ n )  |->  s )  e. 
_V
41, 3dmmpt2 6865 . 2  |-  dom repeatS  =  ( _V  X.  NN0 )
5 df-nel 2665 . . . 4  |-  ( N  e/  NN0  <->  -.  N  e.  NN0 )
65biimpi 194 . . 3  |-  ( N  e/  NN0  ->  -.  N  e.  NN0 )
76intnand 914 . 2  |-  ( N  e/  NN0  ->  -.  ( S  e.  _V  /\  N  e.  NN0 ) )
8 ndmovg 6453 . 2  |-  ( ( dom repeatS  =  ( _V  X.  NN0 )  /\  -.  ( S  e.  _V  /\  N  e.  NN0 )
)  ->  ( S repeatS  N )  =  (/) )
94, 7, 8sylancr 663 1  |-  ( N  e/  NN0  ->  ( S repeatS  N )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    e/ wnel 2663   _Vcvv 3118   (/)c0 3790    |-> cmpt 4511    X. cxp 5003   dom cdm 5005  (class class class)co 6295   0cc0 9504   NN0cn0 10807  ..^cfzo 11804   repeatS creps 12521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-reps 12529
This theorem is referenced by:  repswswrd  12735
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