Help indexed family sets proof

[IntroAnalysis]

Homework Statement

Ai and Bi are indexed families of sets. Prove that Ui (Ai \bigcap Bi) \subseteq (UiAi) \bigcap (UiBi).

Homework Equations

The Attempt at a Solution

Suppose arbitrary x. Let x \in
{x l \foralli\inI(x\inAi\bigcapBi)
This means x \in{x l \foralli\inI(x\inAi)\wedge\foralli\inI(x\inBi).

Homework Statement

This means x \in\neg\existsi\inI(x\notinAi)\wedge\neg\existsi\inI(x\in\notinBi)}
Which is equivalent to: x\in{x l \existsi\inI(x\inAi)\wedge\existsi\inI(x\inBi)}
Therefore, x\in{x l (\bigcupi\inIAi)\bigcap(Ui\inIBi)}
Therefore, is equiv. to Ui\inI(Ai\bigcapBi), then x\in(Ui\inIAi)\bigcap(Ii\inIBi).
Therefore Ui\inI(Ai\bigcapBi)\subseteq(Ui\inIAi)\bigcap(Ii\inIBi).

 

[Stephen Tashi]

The Attempt at a Solution

Suppose arbitrary x. Let x \in
{x l \foralli\inI(x\inAi\bigcapBi)

Are you trying to being by taking an arbitrary x in \bigcup_{i=1}^\infty (A_i \cap B_i) ? If so, why are you applying the "\forall" quantifier to the index?

I suggest beginning this way:

Let x be an arbitrary element of \bigcup_{i=1}^\infty (A_i \cap B_i)

For such an x, we know that there exists an index j such that x \in A_j \cap B_j. This follows from the definition and properties of a union of sets.

This implies x \in A_j and x \in B_j by definition of an intersection of two sets.