Theorem repsundef 14753

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 14751	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2		ovex 7449	. . . 4 ⊢ (0..^𝑛) ∈ V
3	2	mptex 7231	. . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
4	1, 3	dmmpo 8073	. 2 ⊢ dom repeatS = (V × ℕ0)
5		df-nel 3037	. . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
6	5	biimpi 215	. . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
7	6	intnand 487	. 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8		ndmovg 7601	. 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
9	4, 7, 8	sylancr 585	1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∉ wnel 3036  Vcvv 3463  ∅c0 4318   ↦ cmpt 5226   × cxp 5670  dom cdm 5672  (class class class)co 7416  0cc0 11138  ℕ₀cn0 12502  ..^cfzo 13659   repeatS creps 14750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-reps 14751

This theorem is referenced by:  repswswrd  14766