Theorem islvol

Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012)

	Ref	Expression
Hypotheses	lvolset.b	|- B = ( Base ` K )
	lvolset.c	|- C = (
	lvolset.p	|- P = ( LPlanes ` K )
	lvolset.v	|- V = ( LVols ` K )
Assertion	islvol	|- ( K e. A -> ( X e. V <-> ( X e. B /\ E. y e. P y C X ) ) )

Proof

Step	Hyp	Ref	Expression
1		lvolset.b	|- B = ( Base ` K )
2		lvolset.c	|- C = (
3		lvolset.p	|- P = ( LPlanes ` K )
4		lvolset.v	|- V = ( LVols ` K )
5	1 2 3 4	lvolset	|- ( K e. A -> V = { x e. B | E. y e. P y C x } )
6	5	eleq2d	|- ( K e. A -> ( X e. V <-> X e. { x e. B | E. y e. P y C x } ) )
7		breq2	|- ( x = X -> ( y C x <-> y C X ) )
8	7	rexbidv	|- ( x = X -> ( E. y e. P y C x <-> E. y e. P y C X ) )
9	8	elrab	|- ( X e. { x e. B | E. y e. P y C x } <-> ( X e. B /\ E. y e. P y C X ) )
10	6 9	bitrdi	|- ( K e. A -> ( X e. V <-> ( X e. B /\ E. y e. P y C X ) ) )