Theorem repsundef 14745

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 14743	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2		ovex 7447	. . . 4 ⊢ (0..^𝑛) ∈ V
3	2	mptex 7229	. . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
4	1, 3	dmmpo 8069	. 2 ⊢ dom repeatS = (V × ℕ0)
5		df-nel 3042	. . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
6	5	biimpi 215	. . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
7	6	intnand 488	. 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8		ndmovg 7598	. 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
9	4, 7, 8	sylancr 586	1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ∉ wnel 3041  Vcvv 3469  ∅c0 4318   ↦ cmpt 5225   × cxp 5670  dom cdm 5672  (class class class)co 7414  0cc0 11130  ℕ₀cn0 12494  ..^cfzo 13651   repeatS creps 14742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-reps 14743

This theorem is referenced by:  repswswrd  14758