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Theorem pnrmcld 21056
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 21053 . . . 4 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
21simprbi 480 . . 3 (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
32sselda 3583 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
4 eqid 2621 . . . 4 (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)
54elrnmpt 5332 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓))
65adantl 482 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓))
73, 6mpbid 222 1 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wss 3555   cint 4440  cmpt 4673  ran crn 5075  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cn 10964  Clsdccld 20730  Nrmcnrm 21024  PNrmcpnrm 21026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-cnv 5082  df-dm 5084  df-rn 5085  df-iota 5810  df-fv 5855  df-ov 6607  df-pnrm 21033
This theorem is referenced by:  pnrmopn  21057
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