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Theorem iscnrm 17392
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 4026 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . . 5  |-  X  = 
U. J
31, 2syl6eqr 2488 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
43pweqd 3806 . . 3  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
5 oveq1 6091 . . . 4  |-  ( j  =  J  ->  (
jt  x )  =  ( Jt  x ) )
65eleq1d 2504 . . 3  |-  ( j  =  J  ->  (
( jt  x )  e.  Nrm  <->  ( Jt  x )  e.  Nrm ) )
74, 6raleqbidv 2918 . 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 17387 . 2  |- CNrm  =  {
j  e.  Top  |  A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
97, 8elrab2 3096 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ~Pcpw 3801   U.cuni 4017  (class class class)co 6084   ↾t crest 13653   Topctop 16963   Nrmcnrm 17379  CNrmccnrm 17380
This theorem is referenced by:  cnrmtop  17406  iscnrm2  17407  cnrmi  17429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  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 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-cnrm 17387
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