Theorem cnmptid

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

	Ref	Expression
Hypothesis	cnmptid.j	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
Assertion	cnmptid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		cnmptid.j	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
2		mptresid	⊢ ( I ↾ 𝑋 ) = ( 𝑥 ∈ 𝑋 ↦ 𝑥 )
3		idcn	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( I ↾ 𝑋 ) ∈ ( 𝐽 Cn 𝐽 ) )
4	1 3	syl	⊢ ( 𝜑 → ( I ↾ 𝑋 ) ∈ ( 𝐽 Cn 𝐽 ) )
5	2 4	eqeltrrid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) )