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Metamath Proof Explorer

Theorem restcldr

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

Ref Expression
Assertion restcldr 
|- ( ( A e. ( Clsd ` J ) /\ B e. ( Clsd ` ( J |`t A ) ) ) -> B e. ( Clsd ` J ) )

Proof

Step Hyp Ref Expression
1 cldrcl 
 |-  ( A e. ( Clsd ` J ) -> J e. Top )
2 eqid 
 |-  U. J = U. J
3 2 cldss 
 |-  ( A e. ( Clsd ` J ) -> A C_ U. J )
4 2 restcld 
 |-  ( ( J e. Top /\ A C_ U. J ) -> ( B e. ( Clsd ` ( J |`t A ) ) <-> E. v e. ( Clsd ` J ) B = ( v i^i A ) ) )
5 1 3 4 syl2anc 
 |-  ( A e. ( Clsd ` J ) -> ( B e. ( Clsd ` ( J |`t A ) ) <-> E. v e. ( Clsd ` J ) B = ( v i^i A ) ) )
6 incld 
 |-  ( ( v e. ( Clsd ` J ) /\ A e. ( Clsd ` J ) ) -> ( v i^i A ) e. ( Clsd ` J ) )
7 6 ancoms 
 |-  ( ( A e. ( Clsd ` J ) /\ v e. ( Clsd ` J ) ) -> ( v i^i A ) e. ( Clsd ` J ) )
8 eleq1 
 |-  ( B = ( v i^i A ) -> ( B e. ( Clsd ` J ) <-> ( v i^i A ) e. ( Clsd ` J ) ) )
9 7 8 syl5ibrcom 
 |-  ( ( A e. ( Clsd ` J ) /\ v e. ( Clsd ` J ) ) -> ( B = ( v i^i A ) -> B e. ( Clsd ` J ) ) )
10 9 rexlimdva 
 |-  ( A e. ( Clsd ` J ) -> ( E. v e. ( Clsd ` J ) B = ( v i^i A ) -> B e. ( Clsd ` J ) ) )
11 5 10 sylbid 
 |-  ( A e. ( Clsd ` J ) -> ( B e. ( Clsd ` ( J |`t A ) ) -> B e. ( Clsd ` J ) ) )
12 11 imp 
 |-  ( ( A e. ( Clsd ` J ) /\ B e. ( Clsd ` ( J |`t A ) ) ) -> B e. ( Clsd ` J ) )