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Theorem pnrmcld 17070
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  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  E. f  e.  ( J  ^m  NN ) A  =  |^| ran  f )
Distinct variable groups:    A, f    f, J

Proof of Theorem pnrmcld
StepHypRef Expression
1 ispnrm 17067 . . . 4  |-  ( J  e. PNrm 
<->  ( J  e.  Nrm  /\  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) ) )
21simprbi 450 . . 3  |-  ( J  e. PNrm  ->  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
32sselda 3180 . 2  |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  A  e.  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
4 eqid 2283 . . . 4  |-  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f )  =  ( f  e.  ( J  ^m  NN )  |->  |^|
ran  f )
54elrnmpt 4926 . . 3  |-  ( A  e.  ( Clsd `  J
)  ->  ( A  e.  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f )  <->  E. f  e.  ( J  ^m  NN ) A  =  |^| ran  f
) )
65adantl 452 . 2  |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  ( A  e.  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f )  <->  E. f  e.  ( J  ^m  NN ) A  =  |^| ran  f ) )
73, 6mpbid 201 1  |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  E. f  e.  ( J  ^m  NN ) A  =  |^| ran  f )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   |^|cint 3862    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   NNcn 9746   Clsdccld 16753   Nrmcnrm 17038  PNrmcpnrm 17040
This theorem is referenced by:  pnrmopn  17071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861  df-pnrm 17047
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