Theorem sigagenss2 30213

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 30204	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
2	1	3ad2ant3 1084	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
3		simp1 1061	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → ∪ 𝐴 = ∪ 𝐵)
4	3	fveq2d 6195	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigAlgebra‘∪ 𝐴) = (sigAlgebra‘∪ 𝐵))
5	2, 4	eleqtrrd 2704	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴))
6		simp2 1062	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7		sigagenss 30212	. 2 ⊢ (((sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
8	5, 6, 7	syl2anc 693	1 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∪ cuni 4436  ‘cfv 5888  sigAlgebracsiga 30170  sigaGencsigagen 30201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-siga 30171  df-sigagen 30202

This theorem is referenced by:  sxbrsigalem3  30334  sxbrsigalem1  30347  sxbrsigalem2  30348  sxbrsigalem4  30349  sxbrsigalem5  30350  sxbrsiga  30352