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Theorem pnrmcld 17086
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 17083 . . . 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 3193 . 2  |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J
) )  ->  A  e.  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
4 eqid 2296 . . . 4  |-  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f )  =  ( f  e.  ( J  ^m  NN )  |->  |^|
ran  f )
54elrnmpt 4942 . . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   |^|cint 3878    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   NNcn 9762   Clsdccld 16769   Nrmcnrm 17054  PNrmcpnrm 17056
This theorem is referenced by:  pnrmopn  17087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-pnrm 17063
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