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Theorem pnrmcld 17408
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 17405 . . . 4  |-  ( J  e. PNrm 
<->  ( J  e.  Nrm  /\  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) ) )
21simprbi 452 . . 3  |-  ( J  e. PNrm  ->  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
32sselda 3350 . 2  |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  A  e.  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
4 eqid 2438 . . . 4  |-  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f )  =  ( f  e.  ( J  ^m  NN )  |->  |^|
ran  f )
54elrnmpt 5119 . . 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 454 . 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 203 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   |^|cint 4052    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   NNcn 10002   Clsdccld 17082   Nrmcnrm 17376  PNrmcpnrm 17378
This theorem is referenced by:  pnrmopn  17409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464  df-ov 6086  df-pnrm 17385
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