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Metamath Proof Explorer

Theorem cncff

Description: A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008) (Revised by Mario Carneiro, 25-Aug-2014)

		Ref	Expression
	Assertion	cncff	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cncfrss	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → 𝐴 ⊆ ℂ )
2		cncfrss2	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → 𝐵 ⊆ ℂ )
3		elcncf	⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ ) → ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ℝ⁺ ∃ 𝑧 ∈ ℝ⁺ ∀ 𝑤 ∈ 𝐴 ( ( abs ‘ ( 𝑥 − 𝑤 ) ) < 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐹 ‘ 𝑤 ) ) ) < 𝑦 ) ) ) )
4	1 2 3	syl2anc	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ℝ⁺ ∃ 𝑧 ∈ ℝ⁺ ∀ 𝑤 ∈ 𝐴 ( ( abs ‘ ( 𝑥 − 𝑤 ) ) < 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐹 ‘ 𝑤 ) ) ) < 𝑦 ) ) ) )
5	4	ibi	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ℝ⁺ ∃ 𝑧 ∈ ℝ⁺ ∀ 𝑤 ∈ 𝐴 ( ( abs ‘ ( 𝑥 − 𝑤 ) ) < 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐹 ‘ 𝑤 ) ) ) < 𝑦 ) ) )
6	5	simpld	⊢ ( 𝐹 ∈ ( 𝐴 –cn→ 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 )