Theorem limsupresuz2

Description: If the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	limsupresuz2.1	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	limsupresuz2.2	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	limsupresuz2.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
	limsupresuz2.4	⊢ ( 𝜑 → dom 𝐹 ⊆ ℤ )
Assertion	limsupresuz2	⊢ ( 𝜑 → ( lim sup ‘ ( 𝐹 ↾ 𝑍 ) ) = ( lim sup ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		limsupresuz2.1	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2		limsupresuz2.2	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3		limsupresuz2.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
4		limsupresuz2.4	⊢ ( 𝜑 → dom 𝐹 ⊆ ℤ )
5		dmresss	⊢ dom ( 𝐹 ↾ ℝ ) ⊆ dom 𝐹
6	5	a1i	⊢ ( 𝜑 → dom ( 𝐹 ↾ ℝ ) ⊆ dom 𝐹 )
7	6 4	sstrd	⊢ ( 𝜑 → dom ( 𝐹 ↾ ℝ ) ⊆ ℤ )
8	1 2 3 7	limsupresuz	⊢ ( 𝜑 → ( lim sup ‘ ( 𝐹 ↾ 𝑍 ) ) = ( lim sup ‘ 𝐹 ) )