Theorem unielsiga 26691

Description: A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)

Assertion

Ref	Expression
unielsiga	|- ( S e. U. ran sigAlgebra -> U. S e. S )

Proof of Theorem unielsiga

Step	Hyp	Ref	Expression
1		sgon 26687	. 2 |- ( S e. U. ran sigAlgebra -> S e. (sigAlgebra ` U. S ) )
2		baselsiga 26678	. 2 |- ( S e. (sigAlgebra ` U. S ) -> U. S e. S )
3	1, 2	syl 16	1 |- ( S e. U. ran sigAlgebra -> U. S e. S )

Syntax hints:   -> wi 4   e. wcel 1757   U.cuni 4175   ran crn 4925   ` cfv 5502  sigAlgebracsiga 26670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-fv 5510  df-siga 26671

This theorem is referenced by:  mbfmcst  26794  1stmbfm  26795  2ndmbfm  26796  imambfm  26797  mbfmco  26799  br2base  26804  prob01  26916  probfinmeasb  26932