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Theorem pnrmnrm 23064
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 23063 . 2 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽m ℕ) ↦ ran 𝑥)))
21simplbi 496 1 (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  wss 3947   cint 4949  cmpt 5230  ran crn 5676  cfv 6542  (class class class)co 7411  m cmap 8822  cn 12216  Clsdccld 22740  Nrmcnrm 23034  PNrmcpnrm 23036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6494  df-fv 6550  df-ov 7414  df-pnrm 23043
This theorem is referenced by:  pnrmtop  23065
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