Theorem smfpreimale

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded below is in the subspace sigma-algebra induced by its domain. See Proposition 121B (ii) of Fremlin1 p. 35 (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		smfpreimale.s	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
2		smfpreimale.f	⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
3		smfpreimale.d	⊢ 𝐷 = dom 𝐹
4		smfpreimale.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
5	1 3	issmfle	⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) )
6	2 5	mpbid	⊢ ( 𝜑 → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) )
7	6	simp3d	⊢ ( 𝜑 → ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) )
8		breq2	⊢ ( 𝑎 = 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 ↔ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 ) )
9	8	rabbidv	⊢ ( 𝑎 = 𝐴 → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } = { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } )
10	9	eleq1d	⊢ ( 𝑎 = 𝐴 → ( { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ↔ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) ) )
11	10	rspcva	⊢ ( ( 𝐴 ∈ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝑎 } ∈ ( 𝑆 ↾t 𝐷 ) ) → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) )
12	4 7 11	syl2anc	⊢ ( 𝜑 → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ≤ 𝐴 } ∈ ( 𝑆 ↾t 𝐷 ) )