Theorem dstfrvel

Description: Elementhood of preimage maps produced by the "less than or equal to" relation. (Contributed by Thierry Arnoux, 13-Feb-2017)

	Ref	Expression
Hypotheses	dstfrv.1	|- ( ph -> P e. Prob )
	dstfrv.2	|- ( ph -> X e. ( rRndVar ` P ) )
	orvclteel.1	|- ( ph -> A e. RR )
	dstfrvel.1	|- ( ph -> B e. U. dom P )
	dstfrvel.2	|- ( ph -> ( X ` B ) <_ A )
Assertion	dstfrvel	|- ( ph -> B e. ( X oRVC <_ A ) )

Proof

Step	Hyp	Ref	Expression
1		dstfrv.1	|- ( ph -> P e. Prob )
2		dstfrv.2	|- ( ph -> X e. ( rRndVar ` P ) )
3		orvclteel.1	|- ( ph -> A e. RR )
4		dstfrvel.1	|- ( ph -> B e. U. dom P )
5		dstfrvel.2	|- ( ph -> ( X ` B ) <_ A )
6	1 2	rrvvf	|- ( ph -> X : U. dom P --> RR )
7	6 4	ffvelrnd	|- ( ph -> ( X ` B ) e. RR )
8		breq1	|- ( x = ( X ` B ) -> ( x <_ A <-> ( X ` B ) <_ A ) )
9	8	elrab	|- ( ( X ` B ) e. { x e. RR | x <_ A } <-> ( ( X ` B ) e. RR /\ ( X ` B ) <_ A ) )
10	7 5 9	sylanbrc	|- ( ph -> ( X ` B ) e. { x e. RR | x <_ A } )
11	6	ffund	|- ( ph -> Fun X )
12	1 2	rrvdm	|- ( ph -> dom X = U. dom P )
13	4 12	eleqtrrd	|- ( ph -> B e. dom X )
14		fvimacnv	|- ( ( Fun X /\ B e. dom X ) -> ( ( X ` B ) e. { x e. RR | x <_ A } <-> B e. ( `' X " { x e. RR | x <_ A } ) ) )
15	11 13 14	syl2anc	|- ( ph -> ( ( X ` B ) e. { x e. RR | x <_ A } <-> B e. ( `' X " { x e. RR | x <_ A } ) ) )
16	10 15	mpbid	|- ( ph -> B e. ( `' X " { x e. RR | x <_ A } ) )
17	1 2 3	orrvcval4	|- ( ph -> ( X oRVC <_ A ) = ( `' X " { x e. RR | x <_ A } ) )
18	16 17	eleqtrrd	|- ( ph -> B e. ( X oRVC <_ A ) )