Theorem pnrmcld

Description: A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Assertion	pnrmcld	⊢ ( ( 𝐽 ∈ PNrm ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ∃ 𝑓 ∈ ( 𝐽 ↑m ℕ ) 𝐴 = ∩ ran 𝑓 )

Proof

Step	Hyp	Ref	Expression
1		ispnrm	⊢ ( 𝐽 ∈ PNrm ↔ ( 𝐽 ∈ Nrm ∧ ( Clsd ‘ 𝐽 ) ⊆ ran ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) ) )
2	1	simprbi	⊢ ( 𝐽 ∈ PNrm → ( Clsd ‘ 𝐽 ) ⊆ ran ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) )
3	2	sselda	⊢ ( ( 𝐽 ∈ PNrm ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝐴 ∈ ran ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) )
4		eqid	⊢ ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) = ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 )
5	4	elrnmpt	⊢ ( 𝐴 ∈ ( Clsd ‘ 𝐽 ) → ( 𝐴 ∈ ran ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) ↔ ∃ 𝑓 ∈ ( 𝐽 ↑m ℕ ) 𝐴 = ∩ ran 𝑓 ) )
6	5	adantl	⊢ ( ( 𝐽 ∈ PNrm ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 ∈ ran ( 𝑓 ∈ ( 𝐽 ↑m ℕ ) ↦ ∩ ran 𝑓 ) ↔ ∃ 𝑓 ∈ ( 𝐽 ↑m ℕ ) 𝐴 = ∩ ran 𝑓 ) )
7	3 6	mpbid	⊢ ( ( 𝐽 ∈ PNrm ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ∃ 𝑓 ∈ ( 𝐽 ↑m ℕ ) 𝐴 = ∩ ran 𝑓 )