Theorem sigagenid 32119

Description: The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Assertion

Ref	Expression
sigagenid	⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)

Proof of Theorem sigagenid

Step	Hyp	Ref	Expression
1		sgon 32092	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		ssid 3943	. . 3 ⊢ 𝑆 ⊆ 𝑆
3		sigagenss 32117	. . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝑆) ∧ 𝑆 ⊆ 𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
4	1, 2, 3	sylancl 586	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5		sssigagen 32113	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
6	4, 5	eqssd 3938	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  ran crn 5590  ‘cfv 6433  sigAlgebracsiga 32076  sigaGencsigagen 32106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-siga 32077  df-sigagen 32107

This theorem is referenced by: (None)