Theorem repsundef

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)

	Ref	Expression
Assertion	repsundef	⊢ ( 𝑁 ∉ ℕ0 → ( 𝑆 repeatS 𝑁 ) = ∅ )

Proof

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