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Theorem iscnrm2 17082
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 16680 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 eqid 2296 . . . . 5  |-  U. J  =  U. J
32iscnrm 17067 . . . 4  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
43baib 871 . . 3  |-  ( J  e.  Top  ->  ( J  e. CNrm  <->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
51, 4syl 15 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e. CNrm  <->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
6 toponuni 16681 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76pweqd 3643 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ~P X  =  ~P U. J )
87raleqdv 2755 . 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 247 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 176    e. wcel 1696   A.wral 2556   ~Pcpw 3638   U.cuni 3843   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Nrmcnrm 17054  CNrmccnrm 17055
This theorem is referenced by:  restcnrm  17106
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-topon 16655  df-cnrm 17062
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