Theorem sigagenss2 34783

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 34774	. . . 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 2864	. 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 34782	. 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 2145   ⊆ wss 3899  ∪ cuni 4867  ‘cfv 6538  sigAlgebracsiga 34740  sigaGencsigagen 34771

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-siga 34741  df-sigagen 34772

This theorem is used by:  sxbrsigalem3  34904  sxbrsigalem1  34917  sxbrsigalem2  34918  sxbrsigalem4  34919  sxbrsigalem5  34920  sxbrsiga  34922