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Theorem pnrmcld 21475
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 21472 . . . 4 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
21simprbi 491 . . 3 (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
32sselda 3798 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
4 eqid 2799 . . . 4 (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)
54elrnmpt 5576 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓))
65adantl 474 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓))
73, 6mpbid 224 1 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  wss 3769   cint 4667  cmpt 4922  ran crn 5313  cfv 6101  (class class class)co 6878  𝑚 cmap 8095  cn 11312  Clsdccld 21149  Nrmcnrm 21443  PNrmcpnrm 21445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-cnv 5320  df-dm 5322  df-rn 5323  df-iota 6064  df-fv 6109  df-ov 6881  df-pnrm 21452
This theorem is referenced by:  pnrmopn  21476
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