Theorem cnmptc

Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014) (Revised by Mario Carneiro, 22-Aug-2015)

	Ref	Expression
Hypotheses	cnmptid.j	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
	cnmptc.k	⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) )
	cnmptc.p	⊢ ( 𝜑 → 𝑃 ∈ 𝑌 )
Assertion	cnmptc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝑃 ) ∈ ( 𝐽 Cn 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		cnmptid.j	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
2		cnmptc.k	⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) )
3		cnmptc.p	⊢ ( 𝜑 → 𝑃 ∈ 𝑌 )
4		fconstmpt	⊢ ( 𝑋 × { 𝑃 } ) = ( 𝑥 ∈ 𝑋 ↦ 𝑃 )
5		cnconst2	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝑃 ∈ 𝑌 ) → ( 𝑋 × { 𝑃 } ) ∈ ( 𝐽 Cn 𝐾 ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( 𝑋 × { 𝑃 } ) ∈ ( 𝐽 Cn 𝐾 ) )
7	4 6	eqeltrrid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝑃 ) ∈ ( 𝐽 Cn 𝐾 ) )