Theorem constcncfg

Description: A constant function is a continuous function on CC . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	constcncfg.a	⊢ ( 𝜑 → 𝐴 ⊆ ℂ )
	constcncfg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
	constcncfg.c	⊢ ( 𝜑 → 𝐶 ⊆ ℂ )
Assertion	constcncfg	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( 𝐴 –cn→ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		constcncfg.a	⊢ ( 𝜑 → 𝐴 ⊆ ℂ )
2		constcncfg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
3		constcncfg.c	⊢ ( 𝜑 → 𝐶 ⊆ ℂ )
4		cncfmptc	⊢ ( ( 𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ ℂ ∧ 𝐶 ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( 𝐴 –cn→ 𝐶 ) )
5	2 1 3 4	syl3anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( 𝐴 –cn→ 𝐶 ) )