MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscnrm Structured version   Unicode version

Theorem iscnrm 18932
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 4104 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . . 5  |-  X  = 
U. J
31, 2syl6eqr 2493 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
43pweqd 3870 . . 3  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
5 oveq1 6103 . . . 4  |-  ( j  =  J  ->  (
jt  x )  =  ( Jt  x ) )
65eleq1d 2509 . . 3  |-  ( j  =  J  ->  (
( jt  x )  e.  Nrm  <->  ( Jt  x )  e.  Nrm ) )
74, 6raleqbidv 2936 . 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 18927 . 2  |- CNrm  =  {
j  e.  Top  |  A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
97, 8elrab2 3124 1  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   ~Pcpw 3865   U.cuni 4096  (class class class)co 6096   ↾t crest 14364   Topctop 18503   Nrmcnrm 18919  CNrmccnrm 18920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-cnrm 18927
This theorem is referenced by:  cnrmtop  18946  iscnrm2  18947  cnrmi  18969
  Copyright terms: Public domain W3C validator