Theorem salrestss

Description: A sigma-algebra restricted to one of its elements is a subset of the original sigma-algebra. (Contributed by Glauco Siliprandi, 21-Dec-2024)

	Ref	Expression
Hypotheses	salrestss.1	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
	salrestss.2	⊢ ( 𝜑 → 𝐸 ∈ 𝑆 )
Assertion	salrestss	⊢ ( 𝜑 → ( 𝑆 ↾t 𝐸 ) ⊆ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		salrestss.1	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
2		salrestss.2	⊢ ( 𝜑 → 𝐸 ∈ 𝑆 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) )
4	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → 𝑆 ∈ SAlg )
5	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → 𝐸 ∈ 𝑆 )
6		elrest	⊢ ( ( 𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ) → ( 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ↔ ∃ 𝑦 ∈ 𝑆 𝑥 = ( 𝑦 ∩ 𝐸 ) ) )
7	4 5 6	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → ( 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ↔ ∃ 𝑦 ∈ 𝑆 𝑥 = ( 𝑦 ∩ 𝐸 ) ) )
8	3 7	mpbid	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → ∃ 𝑦 ∈ 𝑆 𝑥 = ( 𝑦 ∩ 𝐸 ) )
9		simprr	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑥 = ( 𝑦 ∩ 𝐸 ) ) ) → 𝑥 = ( 𝑦 ∩ 𝐸 ) )
10	1	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → 𝑆 ∈ SAlg )
11		simpr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝑆 )
12	2	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → 𝐸 ∈ 𝑆 )
13	10 11 12	salincld	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑦 ∩ 𝐸 ) ∈ 𝑆 )
14	13	adantrr	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑥 = ( 𝑦 ∩ 𝐸 ) ) ) → ( 𝑦 ∩ 𝐸 ) ∈ 𝑆 )
15	9 14	eqeltrd	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑥 = ( 𝑦 ∩ 𝐸 ) ) ) → 𝑥 ∈ 𝑆 )
16	15	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑥 = ( 𝑦 ∩ 𝐸 ) ) ) → 𝑥 ∈ 𝑆 )
17	8 16	rexlimddv	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ↾t 𝐸 ) ) → 𝑥 ∈ 𝑆 )
18	17	ssd	⊢ ( 𝜑 → ( 𝑆 ↾t 𝐸 ) ⊆ 𝑆 )