Theorem sigagenval

Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016)

	Ref	Expression
Assertion	sigagenval	⊢ ( 𝐴 ∈ 𝑉 → ( sigaGen ‘ 𝐴 ) = ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )

Proof

Step	Hyp	Ref	Expression
1		df-sigagen	⊢ sigaGen = ( 𝑥 ∈ V ↦ ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝑥 ) ∣ 𝑥 ⊆ 𝑠 } )
2	1	a1i	⊢ ( 𝐴 ∈ 𝑉 → sigaGen = ( 𝑥 ∈ V ↦ ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝑥 ) ∣ 𝑥 ⊆ 𝑠 } ) )
3		unieq	⊢ ( 𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴 )
4	3	fveq2d	⊢ ( 𝑥 = 𝐴 → ( sigAlgebra ‘ ∪ 𝑥 ) = ( sigAlgebra ‘ ∪ 𝐴 ) )
5		sseq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠 ) )
6	4 5	rabeqbidv	⊢ ( 𝑥 = 𝐴 → { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝑥 ) ∣ 𝑥 ⊆ 𝑠 } = { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )
7	6	inteqd	⊢ ( 𝑥 = 𝐴 → ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝑥 ) ∣ 𝑥 ⊆ 𝑠 } = ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )
8	7	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴 ) → ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝑥 ) ∣ 𝑥 ⊆ 𝑠 } = ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )
9		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
10		uniexg	⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V )
11		pwsiga	⊢ ( ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) )
12	10 11	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) )
13		pwuni	⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12 13	jctir	⊢ ( 𝐴 ∈ 𝑉 → ( 𝒫 ∪ 𝐴 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴 ) )
15		sseq2	⊢ ( 𝑠 = 𝒫 ∪ 𝐴 → ( 𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴 ) )
16	15	elrab	⊢ ( 𝒫 ∪ 𝐴 ∈ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } ↔ ( 𝒫 ∪ 𝐴 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴 ) )
17	14 16	sylibr	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )
18	17	ne0d	⊢ ( 𝐴 ∈ 𝑉 → { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } ≠ ∅ )
19		intex	⊢ ( { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } ≠ ∅ ↔ ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } ∈ V )
20	18 19	sylib	⊢ ( 𝐴 ∈ 𝑉 → ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } ∈ V )
21	2 8 9 20	fvmptd	⊢ ( 𝐴 ∈ 𝑉 → ( sigaGen ‘ 𝐴 ) = ∩ { 𝑠 ∈ ( sigAlgebra ‘ ∪ 𝐴 ) ∣ 𝐴 ⊆ 𝑠 } )