Theorem smfpimgtxrmpt

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021) (Revised by Glauco Siliprandi, 20-Dec-2024)

	Ref	Expression
Hypotheses	smfpimgtxrmpt.x	|- F/ x ph
	smfpimgtxrmpt.s	|- ( ph -> S e. SAlg )
	smfpimgtxrmpt.b	|- ( ( ph /\ x e. A ) -> B e. V )
	smfpimgtxrmpt.f	|- ( ph -> ( x e. A |-> B ) e. ( SMblFn ` S ) )
	smfpimgtxrmpt.l	|- ( ph -> L e. RR* )
Assertion	smfpimgtxrmpt	|- ( ph -> { x e. A | L < B } e. ( S |`t A ) )

Proof

Step	Hyp	Ref	Expression
1		smfpimgtxrmpt.x	|- F/ x ph
2		smfpimgtxrmpt.s	|- ( ph -> S e. SAlg )
3		smfpimgtxrmpt.b	|- ( ( ph /\ x e. A ) -> B e. V )
4		smfpimgtxrmpt.f	|- ( ph -> ( x e. A |-> B ) e. ( SMblFn ` S ) )
5		smfpimgtxrmpt.l	|- ( ph -> L e. RR* )
6		nfcv	|- F/_ x A
7	1 6 2 3 4 5	smfpimgtxrmptf	|- ( ph -> { x e. A | L < B } e. ( S |`t A ) )