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Theorem repsundef 13718
 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 13492 . . 3 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2 ovex 6841 . . . 4 (0..^𝑛) ∈ V
32mptex 6650 . . 3 (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
41, 3dmmpt2 7408 . 2 dom repeatS = (V × ℕ0)
5 df-nel 3036 . . . 4 (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
65biimpi 206 . . 3 (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
76intnand 1000 . 2 (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8 ndmovg 6982 . 2 ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
94, 7, 8sylancr 698 1 (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ∉ wnel 3035  Vcvv 3340  ∅c0 4058   ↦ cmpt 4881   × cxp 5264  dom cdm 5266  (class class class)co 6813  0cc0 10128  ℕ0cn0 11484  ..^cfzo 12659   repeatS creps 13484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-reps 13492 This theorem is referenced by:  repswswrd  13731
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