Metamath Proof Explorer


Theorem ntrval2

Description: Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007)

Ref Expression
Hypothesis clscld.1
|- X = U. J
Assertion ntrval2
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) )

Proof

Step Hyp Ref Expression
1 clscld.1
 |-  X = U. J
2 difss
 |-  ( X \ S ) C_ X
3 1 clsval2
 |-  ( ( J e. Top /\ ( X \ S ) C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) )
4 2 3 mpan2
 |-  ( J e. Top -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) )
5 4 adantr
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) )
6 dfss4
 |-  ( S C_ X <-> ( X \ ( X \ S ) ) = S )
7 6 biimpi
 |-  ( S C_ X -> ( X \ ( X \ S ) ) = S )
8 7 fveq2d
 |-  ( S C_ X -> ( ( int ` J ) ` ( X \ ( X \ S ) ) ) = ( ( int ` J ) ` S ) )
9 8 adantl
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` ( X \ ( X \ S ) ) ) = ( ( int ` J ) ` S ) )
10 9 difeq2d
 |-  ( ( J e. Top /\ S C_ X ) -> ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) = ( X \ ( ( int ` J ) ` S ) ) )
11 5 10 eqtrd
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` S ) ) )
12 11 difeq2d
 |-  ( ( J e. Top /\ S C_ X ) -> ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) = ( X \ ( X \ ( ( int ` J ) ` S ) ) ) )
13 1 ntropn
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) e. J )
14 1 eltopss
 |-  ( ( J e. Top /\ ( ( int ` J ) ` S ) e. J ) -> ( ( int ` J ) ` S ) C_ X )
15 13 14 syldan
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ X )
16 dfss4
 |-  ( ( ( int ` J ) ` S ) C_ X <-> ( X \ ( X \ ( ( int ` J ) ` S ) ) ) = ( ( int ` J ) ` S ) )
17 15 16 sylib
 |-  ( ( J e. Top /\ S C_ X ) -> ( X \ ( X \ ( ( int ` J ) ` S ) ) ) = ( ( int ` J ) ` S ) )
18 12 17 eqtr2d
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) )