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Metamath Proof Explorer

Theorem iscnrm

Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Hypothesis ist0.1 
|- X = U. J
Assertion iscnrm 
|- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J |`t x ) e. Nrm ) )

Proof

Step Hyp Ref Expression
1 ist0.1 
 |-  X = U. J
2 unieq 
 |-  ( j = J -> U. j = U. J )
3 2 1 eqtr4di 
 |-  ( j = J -> U. j = X )
4 3 pweqd 
 |-  ( j = J -> ~P U. j = ~P X )
5 oveq1 
 |-  ( j = J -> ( j |`t x ) = ( J |`t x ) )
6 5 eleq1d 
 |-  ( j = J -> ( ( j |`t x ) e. Nrm <-> ( J |`t x ) e. Nrm ) )
7 4 6 raleqbidv 
 |-  ( j = J -> ( A. x e. ~P U. j ( j |`t x ) e. Nrm <-> A. x e. ~P X ( J |`t x ) e. Nrm ) )
8 df-cnrm 
 |-  CNrm = { j e. Top | A. x e. ~P U. j ( j |`t x ) e. Nrm }
9 7 8 elrab2 
 |-  ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J |`t x ) e. Nrm ) )