Metamath Proof Explorer

Theorem restcldi

Description: A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Hypothesis restcldi.1 
|- X = U. J
Assertion restcldi 
|- ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> B e. ( Clsd ` ( J |`t A ) ) )

Proof

Step Hyp Ref Expression
1 restcldi.1 
 |-  X = U. J
2 simp2 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> B e. ( Clsd ` J ) )
3 dfss 
 |-  ( B C_ A <-> B = ( B i^i A ) )
4 3 biimpi 
 |-  ( B C_ A -> B = ( B i^i A ) )
5 4 3ad2ant3 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> B = ( B i^i A ) )
6 ineq1 
 |-  ( v = B -> ( v i^i A ) = ( B i^i A ) )
7 6 rspceeqv 
 |-  ( ( B e. ( Clsd ` J ) /\ B = ( B i^i A ) ) -> E. v e. ( Clsd ` J ) B = ( v i^i A ) )
8 2 5 7 syl2anc 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> E. v e. ( Clsd ` J ) B = ( v i^i A ) )
9 cldrcl 
 |-  ( B e. ( Clsd ` J ) -> J e. Top )
10 9 3ad2ant2 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> J e. Top )
11 simp1 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> A C_ X )
12 1 restcld 
 |-  ( ( J e. Top /\ A C_ X ) -> ( B e. ( Clsd ` ( J |`t A ) ) <-> E. v e. ( Clsd ` J ) B = ( v i^i A ) ) )
13 10 11 12 syl2anc 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> ( B e. ( Clsd ` ( J |`t A ) ) <-> E. v e. ( Clsd ` J ) B = ( v i^i A ) ) )
14 8 13 mpbird 
 |-  ( ( A C_ X /\ B e. ( Clsd ` J ) /\ B C_ A ) -> B e. ( Clsd ` ( J |`t A ) ) )