Theorem sigagenss2 34294

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 34285	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
2	1	3ad2ant3 1136	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
3		simp1 1137	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → ∪ 𝐴 = ∪ 𝐵)
4	3	fveq2d 6844	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigAlgebra‘∪ 𝐴) = (sigAlgebra‘∪ 𝐵))
5	2, 4	eleqtrrd 2839	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴))
6		simp2 1138	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7		sigagenss 34293	. 2 ⊢ (((sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
8	5, 6, 7	syl2anc 585	1 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∪ cuni 4850  ‘cfv 6498  sigAlgebracsiga 34252  sigaGencsigagen 34282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-siga 34253  df-sigagen 34283

This theorem is referenced by:  sxbrsigalem3  34416  sxbrsigalem1  34429  sxbrsigalem2  34430  sxbrsigalem4  34431  sxbrsigalem5  34432  sxbrsiga  34434