# Metamath Proof Explorer

## Theorem iscld

Description: The predicate "the class S is a closed set". (Contributed by NM, 2-Oct-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis iscld.1
`|- X = U. J`
Assertion iscld
`|- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )`

### Proof

Step Hyp Ref Expression
1 iscld.1
` |-  X = U. J`
2 1 cldval
` |-  ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )`
3 2 eleq2d
` |-  ( J e. Top -> ( S e. ( Clsd ` J ) <-> S e. { x e. ~P X | ( X \ x ) e. J } ) )`
4 difeq2
` |-  ( x = S -> ( X \ x ) = ( X \ S ) )`
5 4 eleq1d
` |-  ( x = S -> ( ( X \ x ) e. J <-> ( X \ S ) e. J ) )`
6 5 elrab
` |-  ( S e. { x e. ~P X | ( X \ x ) e. J } <-> ( S e. ~P X /\ ( X \ S ) e. J ) )`
7 3 6 syl6bb
` |-  ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S e. ~P X /\ ( X \ S ) e. J ) ) )`
8 1 topopn
` |-  ( J e. Top -> X e. J )`
9 elpw2g
` |-  ( X e. J -> ( S e. ~P X <-> S C_ X ) )`
10 8 9 syl
` |-  ( J e. Top -> ( S e. ~P X <-> S C_ X ) )`
11 10 anbi1d
` |-  ( J e. Top -> ( ( S e. ~P X /\ ( X \ S ) e. J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )`
12 7 11 bitrd
` |-  ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )`