Theorem smfpreimagt

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)

	Ref	Expression
Hypotheses	smfpreimagt.s	|- ( ph -> S e. SAlg )
	smfpreimagt.f	|- ( ph -> F e. ( SMblFn ` S ) )
	smfpreimagt.d	|- D = dom F
	smfpreimagt.a	|- ( ph -> A e. RR )
Assertion	smfpreimagt	|- ( ph -> { x e. D | A < ( F ` x ) } e. ( S |`t D ) )

Proof

Step	Hyp	Ref	Expression
1		smfpreimagt.s	|- ( ph -> S e. SAlg )
2		smfpreimagt.f	|- ( ph -> F e. ( SMblFn ` S ) )
3		smfpreimagt.d	|- D = dom F
4		smfpreimagt.a	|- ( ph -> A e. RR )
5	1 3	issmfgt	|- ( ph -> ( F e. ( SMblFn ` S ) <-> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | a < ( F ` x ) } e. ( S |`t D ) ) ) )
6	2 5	mpbid	|- ( ph -> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | a < ( F ` x ) } e. ( S |`t D ) ) )
7	6	simp3d	|- ( ph -> A. a e. RR { x e. D | a < ( F ` x ) } e. ( S |`t D ) )
8		breq1	|- ( a = A -> ( a < ( F ` x ) <-> A < ( F ` x ) ) )
9	8	rabbidv	|- ( a = A -> { x e. D | a < ( F ` x ) } = { x e. D | A < ( F ` x ) } )
10	9	eleq1d	|- ( a = A -> ( { x e. D | a < ( F ` x ) } e. ( S |`t D ) <-> { x e. D | A < ( F ` x ) } e. ( S |`t D ) ) )
11	10	rspcva	|- ( ( A e. RR /\ A. a e. RR { x e. D | a < ( F ` x ) } e. ( S |`t D ) ) -> { x e. D | A < ( F ` x ) } e. ( S |`t D ) )
12	4 7 11	syl2anc	|- ( ph -> { x e. D | A < ( F ` x ) } e. ( S |`t D ) )