Metamath Proof Explorer

Theorem ssiun2

Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

Ref Expression
Assertion ssiun2 
|- ( x e. A -> B C_ U_ x e. A B )

Proof

Step Hyp Ref Expression
1 rspe 
 |-  ( ( x e. A /\ y e. B ) -> E. x e. A y e. B )
2 1 ex 
 |-  ( x e. A -> ( y e. B -> E. x e. A y e. B ) )
3 eliun 
 |-  ( y e. U_ x e. A B <-> E. x e. A y e. B )
4 2 3 imbitrrdi 
 |-  ( x e. A -> ( y e. B -> y e. U_ x e. A B ) )
5 4 ssrdv 
 |-  ( x e. A -> B C_ U_ x e. A B )