Theorem cncfdmsn

Description: A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	cncfdmsn	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ { 𝐴 } ↦ 𝐵 ) ∈ ( { 𝐴 } –cn→ { 𝐵 } ) )

Proof

Step	Hyp	Ref	Expression
1		cnfdmsn	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ { 𝐴 } ↦ 𝐵 ) ∈ ( 𝒫 { 𝐴 } Cn 𝒫 { 𝐵 } ) )
2		snssi	⊢ ( 𝐴 ∈ ℂ → { 𝐴 } ⊆ ℂ )
3		snssi	⊢ ( 𝐵 ∈ ℂ → { 𝐵 } ⊆ ℂ )
4		eqid	⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
5		eqid	⊢ ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) = ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } )
6		eqid	⊢ ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) = ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } )
7	4 5 6	cncfcn	⊢ ( ( { 𝐴 } ⊆ ℂ ∧ { 𝐵 } ⊆ ℂ ) → ( { 𝐴 } –cn→ { 𝐵 } ) = ( ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) Cn ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) ) )
8	2 3 7	syl2an	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( { 𝐴 } –cn→ { 𝐵 } ) = ( ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) Cn ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) ) )
9	4	cnfldtopon	⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ )
10		simpl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ )
11		restsn2	⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ 𝐴 ∈ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) = 𝒫 { 𝐴 } )
12	9 10 11	sylancr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) = 𝒫 { 𝐴 } )
13		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
14		restsn2	⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ 𝐵 ∈ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) = 𝒫 { 𝐵 } )
15	9 13 14	sylancr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) = 𝒫 { 𝐵 } )
16	12 15	oveq12d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( TopOpen ‘ ℂfld ) ↾t { 𝐴 } ) Cn ( ( TopOpen ‘ ℂfld ) ↾t { 𝐵 } ) ) = ( 𝒫 { 𝐴 } Cn 𝒫 { 𝐵 } ) )
17	8 16	eqtr2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝒫 { 𝐴 } Cn 𝒫 { 𝐵 } ) = ( { 𝐴 } –cn→ { 𝐵 } ) )
18	1 17	eleqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ { 𝐴 } ↦ 𝐵 ) ∈ ( { 𝐴 } –cn→ { 𝐵 } ) )