Theorem unisg 31406

Description: The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
unisg	⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴)

Proof of Theorem unisg

Step	Hyp	Ref	Expression
1		sigagensiga 31404	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))
2		issgon 31386	. . . 4 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴)))
3	1, 2	sylib 220	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴)))
4	3	simprd 498	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (sigaGen‘𝐴))
5	4	eqcomd 2830	1 ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∪ cuni 4841  ran crn 5559  ‘cfv 6358  sigAlgebracsiga 31371  sigaGencsigagen 31401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366  df-siga 31372  df-sigagen 31402

This theorem is referenced by:  unibrsiga  31449  sxsigon  31455  imambfm  31524  cnmbfm  31525  sibf0  31596  sibff  31598  sibfof  31602  sitgclg  31604  orvcval4  31722