Theorem repsundef 14789

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 14787	. . 3 ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
2		ovex 7438	. . . 4 ⊢ (0..^𝑛) ∈ V
3	2	mptex 7215	. . 3 ⊢ (𝑥 ∈ (0..^𝑛) ↦ 𝑠) ∈ V
4	1, 3	dmmpo 8070	. 2 ⊢ dom repeatS = (V × ℕ0)
5		df-nel 3037	. . . 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 7590	. 2 ⊢ ((dom repeatS = (V × ℕ0) ∧ ¬ (𝑆 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑆 repeatS 𝑁) = ∅)
9	4, 7, 8	sylancr 587	1 ⊢ (𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∉ wnel 3036  Vcvv 3459  ∅c0 4308   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  (class class class)co 7405  0cc0 11129  ℕ₀cn0 12501  ..^cfzo 13671   repeatS creps 14786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-reps 14787

This theorem is referenced by:  repswswrd  14802