Theorem sigagenss2 34605

Description: Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)

Assertion

Ref	Expression
sigagenss2	⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Proof of Theorem sigagenss2

Step	Hyp	Ref	Expression
1		sigagensiga 34596	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
2	1	3ad2ant3 1153	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
3		simp1 1154	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → ∪ 𝐴 = ∪ 𝐵)
4	3	fveq2d 6889	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigAlgebra‘∪ 𝐴) = (sigAlgebra‘∪ 𝐵))
5	2, 4	eleqtrrd 2868	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴))
6		simp2 1155	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7		sigagenss 34604	. 2 ⊢ (((sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
8	5, 6, 7	syl2anc 596	1 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6540  sigAlgebracsiga 34562  sigaGencsigagen 34593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-siga 34563  df-sigagen 34594

This theorem is used by:  sxbrsigalem3  34727  sxbrsigalem1  34740  sxbrsigalem2  34741  sxbrsigalem4  34742  sxbrsigalem5  34743  sxbrsiga  34745