Theorem smfpreimage

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

	Ref	Expression
Hypotheses	smfpreimage.s	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
	smfpreimage.f	⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
	smfpreimage.d	⊢ 𝐷 = dom 𝐹
	smfpreimage.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	smfpreimage	⊢ ( 𝜑 → { 𝑥 ∈ 𝐷 ∣ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		smfpreimage.s	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
2		smfpreimage.f	⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
3		smfpreimage.d	⊢ 𝐷 = dom 𝐹
4		smfpreimage.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
5	1 3	issmfge	⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) )
6	2 5	mpbid	⊢ ( 𝜑 → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) )
7	6	simp3d	⊢ ( 𝜑 → ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) )
8		breq1	⊢ ( 𝑎 = 𝐴 → ( 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) ↔ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) ) )
9	8	rabbidv	⊢ ( 𝑎 = 𝐴 → { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐷 ∣ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) } )
10	9	eleq1d	⊢ ( 𝑎 = 𝐴 → ( { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ↔ { 𝑥 ∈ 𝐷 ∣ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) )
11	10	rspcva	⊢ ( ( 𝐴 ∈ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) → { 𝑥 ∈ 𝐷 ∣ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) )
12	4 7 11	syl2anc	⊢ ( 𝜑 → { 𝑥 ∈ 𝐷 ∣ 𝐴 ≤ ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) )