Theorem inintabss

Description: Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020)

	Ref	Expression
Assertion	inintabss	|- ( A i^i |^| { x | ph } ) C_ |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) }

Proof

Step	Hyp	Ref	Expression
1		ax-1	|- ( u e. A -> ( E. x ph -> u e. A ) )
2	1	anim1i	|- ( ( u e. A /\ A. x ( ph -> u e. x ) ) -> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) )
3		elinintab	|- ( u e. ( A i^i |^| { x | ph } ) <-> ( u e. A /\ A. x ( ph -> u e. x ) ) )
4		elinintrab	|- ( u e. _V -> ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } <-> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) ) )
5	4	elv	|- ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } <-> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) )
6	2 3 5	3imtr4i	|- ( u e. ( A i^i |^| { x | ph } ) -> u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } )
7	6	ssriv	|- ( A i^i |^| { x | ph } ) C_ |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) }