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Theorem ispnrm 17395
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 5720 . . 3  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
2 oveq1 6080 . . . . 5  |-  ( j  =  J  ->  (
j  ^m  NN )  =  ( J  ^m  NN ) )
32mpteq1d 4282 . . . 4  |-  ( j  =  J  ->  (
f  e.  ( j  ^m  NN )  |->  |^|
ran  f )  =  ( f  e.  ( J  ^m  NN ) 
|->  |^| ran  f ) )
43rneqd 5089 . . 3  |-  ( j  =  J  ->  ran  ( f  e.  ( j  ^m  NN ) 
|->  |^| ran  f )  =  ran  ( f  e.  ( J  ^m  NN )  |->  |^| ran  f ) )
51, 4sseq12d 3369 . 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 17375 . 2  |- PNrm  =  {
j  e.  Nrm  | 
( Clsd `  j )  C_ 
ran  ( f  e.  ( j  ^m  NN )  |->  |^| ran  f ) }
75, 6elrab2 3086 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 1652    e. wcel 1725    C_ wss 3312   |^|cint 4042    e. cmpt 4258   ran crn 4871   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   NNcn 9992   Clsdccld 17072   Nrmcnrm 17366  PNrmcpnrm 17368
This theorem is referenced by:  pnrmnrm  17396  pnrmcld  17398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-dm 4880  df-rn 4881  df-iota 5410  df-fv 5454  df-ov 6076  df-pnrm 17375
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