Theorem repsundef 14698

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.

Step	Hyp	Ref	Expression
1		df-reps 14696	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2		ovex 7393	. . . 4 ⊢ (0..^𝑛) ∈ V
3	2	mptex 7171	. . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
4	1, 3	dmmpo 8017	. 2 ⊢ dom repeatS = (V × ℕ0)
5		df-nel 3038	. . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
6	5	biimpi 216	. . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
7	6	intnand 488	. 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8		ndmovg 7543	. 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
9	4, 7, 8	sylancr 588	1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037  Vcvv 3441  ∅c0 4286   ↦ cmpt 5180   × cxp 5623  dom cdm 5625  (class class class)co 7360  0cc0 11030  ℕ₀cn0 12405  ..^cfzo 13574   repeatS creps 14695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-reps 14696

This theorem is referenced by:  repswswrd  14711