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Metamath Proof Explorer

Theorem idcncfg

Description: The identity function is a continuous function on CC . (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses idcncfg.a ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
idcncfg.b ⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
Assertion idcncfg ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 idcncfg.a ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2 idcncfg.b ⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
3 cncfmptid ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )