Theorem sgon 34105

Description: A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)

Assertion

Ref	Expression
sgon	⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))

Proof of Theorem sgon

Step	Hyp	Ref	Expression
1		eqid 2735	. 2 ⊢ ∪ 𝑆 = ∪ 𝑆
2		issgon 34104	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆))
3	2	biimpri 228	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆) → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
4	1, 3	mpan2 691	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))

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

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

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-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

This theorem is referenced by:  elsigass  34106  isrnsigau  34108  unielsiga  34109  sigagenid  34132  1stmbfm  34242  2ndmbfm  34243  unveldomd  34397  probmeasb  34412