Theorem sigagenid 33149

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 33122	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		ssid 4005	. . 3 ⊢ 𝑆 ⊆ 𝑆
3		sigagenss 33147	. . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝑆) ∧ 𝑆 ⊆ 𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
4	1, 2, 3	sylancl 587	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5		sssigagen 33143	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
6	4, 5	eqssd 4000	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ⊆ wss 3949  ∪ cuni 4909  ran crn 5678  ‘cfv 6544  sigAlgebracsiga 33106  sigaGencsigagen 33136

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-siga 33107  df-sigagen 33137

This theorem is referenced by: (None)