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Theorem repsundef 13369
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 ∉ ℕ0 → (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 13161 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2 ovex 6577 . . . 4 (0..^𝑛) ∈ V
32mptex 6390 . . 3 (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
41, 3dmmpt2 7129 . 2 dom repeatS = (V × ℕ0)
5 df-nel 2783 . . . 4 (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
65biimpi 205 . . 3 (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
76intnand 953 . 2 (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8 ndmovg 6715 . 2 ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
94, 7, 8sylancr 694 1 (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wnel 2781  Vcvv 3173  c0 3874  cmpt 4643   × cxp 5036  dom cdm 5038  (class class class)co 6549  0cc0 9815  0cn0 11169  ..^cfzo 12334   repeatS creps 13153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-reps 13161
This theorem is referenced by:  repswswrd  13382
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