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Theorem iscnrm2 17404
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
iscnrm2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e. CNrm  <->  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
Distinct variable groups:    x, J    x, X

Proof of Theorem iscnrm2
StepHypRef Expression
1 topontop 16993 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 eqid 2438 . . . . 5  |-  U. J  =  U. J
32iscnrm 17389 . . . 4  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
43baib 873 . . 3  |-  ( J  e.  Top  ->  ( J  e. CNrm  <->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
51, 4syl 16 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e. CNrm  <->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
6 toponuni 16994 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76pweqd 3806 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ~P X  =  ~P U. J )
87raleqdv 2912 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A. x  e.  ~P  X
( Jt  x )  e.  Nrm  <->  A. x  e.  ~P  U. J
( Jt  x )  e.  Nrm ) )
95, 8bitr4d 249 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e. CNrm  <->  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   A.wral 2707   ~Pcpw 3801   U.cuni 4017   ` cfv 5456  (class class class)co 6083   ↾t crest 13650   Topctop 16960  TopOnctopon 16961   Nrmcnrm 17376  CNrmccnrm 17377
This theorem is referenced by:  restcnrm  17428
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-13 1728  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-pow 4379  ax-pr 4405  ax-un 4703
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-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-topon 16968  df-cnrm 17384
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