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Theorem iscnrm 17051
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
ist0.1  |-  X  = 
U. J
Assertion
Ref Expression
iscnrm  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
Distinct variable groups:    x, J    x, X

Proof of Theorem iscnrm
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 unieq 3836 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . . 5  |-  X  = 
U. J
31, 2syl6eqr 2333 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
43pweqd 3630 . . 3  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
5 oveq1 5865 . . . 4  |-  ( j  =  J  ->  (
jt  x )  =  ( Jt  x ) )
65eleq1d 2349 . . 3  |-  ( j  =  J  ->  (
( jt  x )  e.  Nrm  <->  ( Jt  x )  e.  Nrm ) )
74, 6raleqbidv 2748 . 2  |-  ( j  =  J  ->  ( A. x  e.  ~P  U. j ( jt  x )  e.  Nrm  <->  A. x  e.  ~P  X ( Jt  x )  e.  Nrm )
)
8 df-cnrm 17046 . 2  |- CNrm  =  {
j  e.  Top  |  A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
97, 8elrab2 2925 1  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ~Pcpw 3625   U.cuni 3827  (class class class)co 5858   ↾t crest 13325   Topctop 16631   Nrmcnrm 17038  CNrmccnrm 17039
This theorem is referenced by:  cnrmtop  17065  iscnrm2  17066  cnrmi  17088
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cnrm 17046
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