Theorem sgon 34091

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 2729	. 2 ⊢ ∪ 𝑆 = ∪ 𝑆
2		issgon 34090	. . 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 1540   ∈ wcel 2109  ∪ cuni 4858  ran crn 5620  ‘cfv 6482  sigAlgebracsiga 34075

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-siga 34076

This theorem is referenced by:  elsigass  34092  isrnsigau  34094  unielsiga  34095  sigagenid  34118  1stmbfm  34228  2ndmbfm  34229  unveldomd  34383  probmeasb  34398