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Theorem pnrmcld 22086
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 ∈ (𝐽m ℕ)𝐴 = ran 𝑓)
Distinct variable groups:   𝐴,𝑓   𝑓,𝐽

Proof of Theorem pnrmcld
StepHypRef Expression
1 ispnrm 22083 . . . 4 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
21simprbi 500 . . 3 (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
32sselda 3875 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
4 eqid 2738 . . . 4 (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)
54elrnmpt 5793 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽m ℕ)𝐴 = ran 𝑓))
65adantl 485 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽m ℕ)𝐴 = ran 𝑓))
73, 6mpbid 235 1 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽m ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  wrex 3054  wss 3841   cint 4833  cmpt 5107  ran crn 5520  cfv 6333  (class class class)co 7164  m cmap 8430  cn 11709  Clsdccld 21760  Nrmcnrm 22054  PNrmcpnrm 22056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6291  df-fv 6341  df-ov 7167  df-pnrm 22063
This theorem is referenced by:  pnrmopn  22087
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