Metamath Proof Explorer


Theorem idcn

Description: A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006) (Proof shortened by Mario Carneiro, 21-Mar-2015)

Ref Expression
Assertion idcn
|- ( J e. ( TopOn ` X ) -> ( _I |` X ) e. ( J Cn J ) )

Proof

Step Hyp Ref Expression
1 ssid
 |-  J C_ J
2 ssidcn
 |-  ( ( J e. ( TopOn ` X ) /\ J e. ( TopOn ` X ) ) -> ( ( _I |` X ) e. ( J Cn J ) <-> J C_ J ) )
3 2 anidms
 |-  ( J e. ( TopOn ` X ) -> ( ( _I |` X ) e. ( J Cn J ) <-> J C_ J ) )
4 1 3 mpbiri
 |-  ( J e. ( TopOn ` X ) -> ( _I |` X ) e. ( J Cn J ) )