Theorem pnrmcld 22493

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

Assertion

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

Distinct variable groups:   𝐴,𝑓   𝑓,𝐽

Proof of Theorem pnrmcld

Step	Hyp	Ref	Expression
1		ispnrm 22490	. . . 4 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
2	1	simprbi 497	. . 3 ⊢ (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
3	2	sselda 3921	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
4		eqid 2738	. . . 4 ⊢ (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)
5	4	elrnmpt 5865	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓))
6	5	adantl 482	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓))
7	3, 6	mpbid 231	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887  ∩ cint 4879   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  ℕcn 11973  Clsdccld 22167  Nrmcnrm 22461  PNrmcpnrm 22463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-cnv 5597  df-dm 5599  df-rn 5600  df-iota 6391  df-fv 6441  df-ov 7278  df-pnrm 22470

This theorem is referenced by:  pnrmopn  22494