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Theorem iscnrm 17349
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 3992 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . . 5  |-  X  = 
U. J
31, 2syl6eqr 2462 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
43pweqd 3772 . . 3  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
5 oveq1 6055 . . . 4  |-  ( j  =  J  ->  (
jt  x )  =  ( Jt  x ) )
65eleq1d 2478 . . 3  |-  ( j  =  J  ->  (
( jt  x )  e.  Nrm  <->  ( Jt  x )  e.  Nrm ) )
74, 6raleqbidv 2884 . 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 17344 . 2  |- CNrm  =  {
j  e.  Top  |  A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
97, 8elrab2 3062 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   ~Pcpw 3767   U.cuni 3983  (class class class)co 6048   ↾t crest 13611   Topctop 16921   Nrmcnrm 17336  CNrmccnrm 17337
This theorem is referenced by:  cnrmtop  17363  iscnrm2  17364  cnrmi  17386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-cnrm 17344
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