This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ntrfval

Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis cldval.1 
|- X = U. J
Assertion ntrfval 
|- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )

Proof

Step Hyp Ref Expression
1 cldval.1 
 |-  X = U. J
2 1 topopn 
 |-  ( J e. Top -> X e. J )
3 pwexg 
 |-  ( X e. J -> ~P X e. _V )
4 mptexg 
 |-  ( ~P X e. _V -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
5 2 3 4 3syl 
 |-  ( J e. Top -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
6 unieq 
 |-  ( j = J -> U. j = U. J )
7 6 1 eqtr4di 
 |-  ( j = J -> U. j = X )
8 7 pweqd 
 |-  ( j = J -> ~P U. j = ~P X )
9 ineq1 
 |-  ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10 9 unieqd 
 |-  ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11 8 10 mpteq12dv 
 |-  ( j = J -> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
12 df-ntr 
 |-  int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13 11 12 fvmptg 
 |-  ( ( J e. Top /\ ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V ) -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
14 5 13 mpdan 
 |-  ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )