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Theorem iscnrm 19950
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 4259 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . . 5  |-  X  = 
U. J
31, 2syl6eqr 2516 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
43pweqd 4020 . . 3  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
5 oveq1 6303 . . . 4  |-  ( j  =  J  ->  (
jt  x )  =  ( Jt  x ) )
65eleq1d 2526 . . 3  |-  ( j  =  J  ->  (
( jt  x )  e.  Nrm  <->  ( Jt  x )  e.  Nrm ) )
74, 6raleqbidv 3068 . 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 19945 . 2  |- CNrm  =  {
j  e.  Top  |  A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
97, 8elrab2 3259 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 1395    e. wcel 1819   A.wral 2807   ~Pcpw 4015   U.cuni 4251  (class class class)co 6296   ↾t crest 14837   Topctop 19520   Nrmcnrm 19937  CNrmccnrm 19938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cnrm 19945
This theorem is referenced by:  cnrmtop  19964  iscnrm2  19965  cnrmi  19987
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