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Theorem pnrmnrm 22728
Description: A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
pnrmnrm (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Proof of Theorem pnrmnrm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ispnrm 22727 . 2 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽m ℕ) ↦ ran 𝑥)))
21simplbi 498 1 (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3913   cint 4912  cmpt 5193  ran crn 5639  cfv 6501  (class class class)co 7362  m cmap 8772  cn 12162  Clsdccld 22404  Nrmcnrm 22698  PNrmcpnrm 22700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-cnv 5646  df-dm 5648  df-rn 5649  df-iota 6453  df-fv 6509  df-ov 7365  df-pnrm 22707
This theorem is referenced by:  pnrmtop  22729
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