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Theorem pnrmnrm 22714
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 22713 . 2 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑥 ∈ (𝐽m ℕ) ↦ ran 𝑥)))
21simplbi 499 1 (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3914   cint 4911  cmpt 5192  ran crn 5638  cfv 6500  (class class class)co 7361  m cmap 8771  cn 12161  Clsdccld 22390  Nrmcnrm 22684  PNrmcpnrm 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-cnv 5645  df-dm 5647  df-rn 5648  df-iota 6452  df-fv 6508  df-ov 7364  df-pnrm 22693
This theorem is referenced by:  pnrmtop  22715
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