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Metamath Proof Explorer

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 } ) )