Theorem sigagenid 34152

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 34125	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		ssid 4006	. . 3 ⊢ 𝑆 ⊆ 𝑆
3		sigagenss 34150	. . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝑆) ∧ 𝑆 ⊆ 𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
4	1, 2, 3	sylancl 586	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5		sssigagen 34146	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
6	4, 5	eqssd 4001	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686  ‘cfv 6561  sigAlgebracsiga 34109  sigaGencsigagen 34139

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-siga 34110  df-sigagen 34140

This theorem is referenced by: (None)