Theorem unielsiga 28361

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 28357	. 2 |- ( S e. U. ran sigAlgebra -> S e. (sigAlgebra ` U. S ) )
2		baselsiga 28348	. 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 1823   U.cuni 4235   ran crn 4989   ` cfv 5570  sigAlgebracsiga 28340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-siga 28341

This theorem is referenced by:  mbfmcst  28470  1stmbfm  28471  2ndmbfm  28472  imambfm  28473  mbfmco  28475  br2base  28480  prob01  28619  probfinmeasb  28635