Theorem sigagenss2 33994

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 33985	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
2	1	3ad2ant3 1132	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐵))
3		simp1 1133	. . . 4 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → ∪ 𝐴 = ∪ 𝐵)
4	3	fveq2d 6895	. . 3 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigAlgebra‘∪ 𝐴) = (sigAlgebra‘∪ 𝐵))
5	2, 4	eleqtrrd 2829	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴))
6		simp2 1134	. 2 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7		sigagenss 33993	. 2 ⊢ (((sigaGen‘𝐵) ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
8	5, 6, 7	syl2anc 582	1 ⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ⊆ wss 3947  ∪ cuni 4906  ‘cfv 6544  sigAlgebracsiga 33952  sigaGencsigagen 33982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6496  df-fun 6546  df-fv 6552  df-siga 33953  df-sigagen 33983

This theorem is referenced by:  sxbrsigalem3  34117  sxbrsigalem1  34130  sxbrsigalem2  34131  sxbrsigalem4  34132  sxbrsigalem5  34133  sxbrsiga  34135