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Theorem ispnrm 17067
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 5525 . . 3  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
2 oveq1 5865 . . . . 5  |-  ( j  =  J  ->  (
j  ^m  NN )  =  ( J  ^m  NN ) )
3 mpteq1 4100 . . . . 5  |-  ( ( j  ^m  NN )  =  ( J  ^m  NN )  ->  ( f  e.  ( j  ^m  NN )  |->  |^| ran  f )  =  ( f  e.  ( J  ^m  NN )  |->  |^|
ran  f ) )
42, 3syl 15 . . . 4  |-  ( j  =  J  ->  (
f  e.  ( j  ^m  NN )  |->  |^|
ran  f )  =  ( f  e.  ( J  ^m  NN ) 
|->  |^| ran  f ) )
54rneqd 4906 . . 3  |-  ( j  =  J  ->  ran  ( f  e.  ( j  ^m  NN ) 
|->  |^| ran  f )  =  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
61, 5sseq12d 3207 . 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 ) ) )
7 df-pnrm 17047 . 2  |- PNrm  =  {
j  e.  Nrm  | 
( Clsd `  j )  C_ 
ran  ( f  e.  ( j  ^m  NN )  |->  |^| ran  f ) }
86, 7elrab2 2925 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    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:  pnrmnrm  17068  pnrmcld  17070
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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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