Theorem smfpimltxrmpt

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below 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	smfpimltxrmpt.x	$$⊢Ⅎx\phi$$
	smfpimltxrmpt.s	$$⊢\phi \to S\in \mathrm{SAlg}$$
	smfpimltxrmpt.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	smfpimltxrmpt.f	$$⊢\phi \to \left(x\in A⟼B\right)\in \mathrm{SMblFn}\left(S\right)$$
	smfpimltxrmpt.r	$$⊢\phi \to R\in {ℝ}^{*}$$
Assertion	smfpimltxrmpt	$$⊢\phi \to \left\{x\in A|B<R\right\}\in \left(S{\upharpoonright }_{𝑡}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		smfpimltxrmpt.x	$$⊢Ⅎx\phi$$
2		smfpimltxrmpt.s	$$⊢\phi \to S\in \mathrm{SAlg}$$
3		smfpimltxrmpt.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
4		smfpimltxrmpt.f	$$⊢\phi \to \left(x\in A⟼B\right)\in \mathrm{SMblFn}\left(S\right)$$
5		smfpimltxrmpt.r	$$⊢\phi \to R\in {ℝ}^{*}$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
7	1 6 2 3 4 5	smfpimltxrmptf	$$⊢\phi \to \left\{x\in A|B<R\right\}\in \left(S{\upharpoonright }_{𝑡}A\right)$$