Theorem unisg 34124

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 34122	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))
2		issgon 34104	. . . 4 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴)))
3	1, 2	sylib 218	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴)))
4	3	simprd 495	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (sigaGen‘𝐴))
5	4	eqcomd 2741	1 ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∪ cuni 4912  ran crn 5690  ‘cfv 6563  sigAlgebracsiga 34089  sigaGencsigagen 34119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-siga 34090  df-sigagen 34120

This theorem is referenced by:  unibrsiga  34167  sxsigon  34173  imambfm  34244  cnmbfm  34245  sibf0  34316  sibff  34318  sibfof  34322  sitgclg  34324  orvcval4  34442