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Theorem pnrmnrm 17404
Description: A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
pnrmnrm  |-  ( J  e. PNrm  ->  J  e.  Nrm )

Proof of Theorem pnrmnrm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ispnrm 17403 . 2  |-  ( J  e. PNrm 
<->  ( J  e.  Nrm  /\  ( Clsd `  J
)  C_  ran  ( x  e.  ( J  ^m  NN )  |->  |^| ran  x ) ) )
21simplbi 447 1  |-  ( J  e. PNrm  ->  J  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3320   |^|cint 4050    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   NNcn 10000   Clsdccld 17080   Nrmcnrm 17374  PNrmcpnrm 17376
This theorem is referenced by:  pnrmtop  17405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-pnrm 17383
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