Metamath Proof Explorer


Theorem issal

Description: Express the predicate " S is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion issal ( 𝑆𝑉 → ( 𝑆 ∈ SAlg ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) ) )

Proof

Step Hyp Ref Expression
1 eleq2 ( 𝑥 = 𝑆 → ( ∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆 ) )
2 id ( 𝑥 = 𝑆𝑥 = 𝑆 )
3 unieq ( 𝑥 = 𝑆 𝑥 = 𝑆 )
4 3 difeq1d ( 𝑥 = 𝑆 → ( 𝑥𝑦 ) = ( 𝑆𝑦 ) )
5 4 2 eleq12d ( 𝑥 = 𝑆 → ( ( 𝑥𝑦 ) ∈ 𝑥 ↔ ( 𝑆𝑦 ) ∈ 𝑆 ) )
6 2 5 raleqbidv ( 𝑥 = 𝑆 → ( ∀ 𝑦𝑥 ( 𝑥𝑦 ) ∈ 𝑥 ↔ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ) )
7 pweq ( 𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆 )
8 eleq2 ( 𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆 ) )
9 8 imbi2d ( 𝑥 = 𝑆 → ( ( 𝑦 ≼ ω → 𝑦𝑥 ) ↔ ( 𝑦 ≼ ω → 𝑦𝑆 ) ) )
10 7 9 raleqbidv ( 𝑥 = 𝑆 → ( ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → 𝑦𝑥 ) ↔ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) )
11 1 6 10 3anbi123d ( 𝑥 = 𝑆 → ( ( ∅ ∈ 𝑥 ∧ ∀ 𝑦𝑥 ( 𝑥𝑦 ) ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → 𝑦𝑥 ) ) ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) ) )
12 df-salg SAlg = { 𝑥 ∣ ( ∅ ∈ 𝑥 ∧ ∀ 𝑦𝑥 ( 𝑥𝑦 ) ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → 𝑦𝑥 ) ) }
13 11 12 elab2g ( 𝑆𝑉 → ( 𝑆 ∈ SAlg ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) ) )