Theorem repsundef 14728

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 14726	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2		ovex 7392	. . . 4 ⊢ (0..^𝑛) ∈ V
3	2	mptex 7170	. . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
4	1, 3	dmmpo 8015	. 2 ⊢ dom repeatS = (V × ℕ0)
5		df-nel 3041	. . . 4 ⊢ (𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0)
6	5	biimpi 218	. . 3 ⊢ (𝑁 ∉ ℕ0 → ¬ 𝑁 ∈ ℕ0)
7	6	intnand 490	. 2 ⊢ (𝑁 ∉ ℕ0 → ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0))
8		ndmovg 7542	. 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
9	4, 7, 8	sylancr 594	1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ∉ wnel 3040  Vcvv 3433  ∅c0 4263   ↦ cmpt 5155   × cxp 5618  dom cdm 5620  (class class class)co 7359  0cc0 11034  ℕ₀cn0 12432  ..^cfzo 13603   repeatS creps 14725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-reps 14726

This theorem is referenced by:  repswswrd  14741