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Theorem ispnrm 17325
Description: The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
ispnrm  |-  ( J  e. PNrm 
<->  ( J  e.  Nrm  /\  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) ) )
Distinct variable group:    f, J

Proof of Theorem ispnrm
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5668 . . 3  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
2 oveq1 6027 . . . . 5  |-  ( j  =  J  ->  (
j  ^m  NN )  =  ( J  ^m  NN ) )
32mpteq1d 4231 . . . 4  |-  ( j  =  J  ->  (
f  e.  ( j  ^m  NN )  |->  |^|
ran  f )  =  ( f  e.  ( J  ^m  NN ) 
|->  |^| ran  f ) )
43rneqd 5037 . . 3  |-  ( j  =  J  ->  ran  ( f  e.  ( j  ^m  NN ) 
|->  |^| ran  f )  =  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
51, 4sseq12d 3320 . 2  |-  ( j  =  J  ->  (
( Clsd `  j )  C_ 
ran  ( f  e.  ( j  ^m  NN )  |->  |^| ran  f )  <-> 
( Clsd `  J )  C_ 
ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) ) )
6 df-pnrm 17305 . 2  |- PNrm  =  {
j  e.  Nrm  | 
( Clsd `  j )  C_ 
ran  ( f  e.  ( j  ^m  NN )  |->  |^| ran  f ) }
75, 6elrab2 3037 1  |-  ( J  e. PNrm 
<->  ( J  e.  Nrm  /\  ( Clsd `  J
)  C_  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   |^|cint 3992    e. cmpt 4207   ran crn 4819   ` cfv 5394  (class class class)co 6020    ^m cmap 6954   NNcn 9932   Clsdccld 17003   Nrmcnrm 17296  PNrmcpnrm 17298
This theorem is referenced by:  pnrmnrm  17326  pnrmcld  17328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-cnv 4826  df-dm 4828  df-rn 4829  df-iota 5358  df-fv 5402  df-ov 6023  df-pnrm 17305
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