MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnrmcld Structured version   Visualization version   GIF version

Theorem pnrmcld 21369
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 21366 . . . 4 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
21simprbi 483 . . 3 (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
32sselda 3745 . 2 ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
4 eqid 2761 . . . 4 (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)
54elrnmpt 5528 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓))
65adantl 473 . 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 383   = wceq 1632  wcel 2140  wrex 3052  wss 3716   cint 4628  cmpt 4882  ran crn 5268  cfv 6050  (class class class)co 6815  𝑚 cmap 8026  cn 11233  Clsdccld 21043  Nrmcnrm 21337  PNrmcpnrm 21339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-cnv 5275  df-dm 5277  df-rn 5278  df-iota 6013  df-fv 6058  df-ov 6818  df-pnrm 21346
This theorem is referenced by:  pnrmopn  21370
  Copyright terms: Public domain W3C validator