Theorem sigagensiga 30204

Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
sigagensiga	|- ( A e. V -> (sigaGen ` A ) e. (sigAlgebra ` U. A ) )

Proof of Theorem sigagensiga

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sigagenval 30203	. 2 |- ( A e. V -> (sigaGen ` A ) = |^| { s e. (sigAlgebra ` U. A ) | A C_ s } )
2		fvex 6201	. . . . 5 |- (sigaGen ` A ) e. _V
3	1, 2	syl6eqelr 2710	. . . 4 |- ( A e. V -> |^| { s e. (sigAlgebra ` U. A ) | A C_ s } e. _V )
4		intex 4820	. . . 4 |- ( { s e. (sigAlgebra ` U. A ) | A C_ s } =/= (/) <-> |^| { s e. (sigAlgebra ` U. A ) | A C_ s } e. _V )
5	3, 4	sylibr 224	. . 3 |- ( A e. V -> { s e. (sigAlgebra ` U. A ) | A C_ s } =/= (/) )
6		ssrab2 3687	. . . . 5 |- { s e. (sigAlgebra ` U. A ) | A C_ s } C_ (sigAlgebra ` U. A )
7	6	a1i 11	. . . 4 |- ( A e. V -> { s e. (sigAlgebra ` U. A ) | A C_ s } C_ (sigAlgebra ` U. A ) )
8		fvex 6201	. . . . 5 |- (sigAlgebra ` U. A ) e. _V
9	8	elpw2 4828	. . . 4 |- ( { s e. (sigAlgebra ` U. A ) | A C_ s } e. ~P (sigAlgebra ` U. A ) <-> { s e. (sigAlgebra ` U. A ) | A C_ s } C_ (sigAlgebra ` U. A ) )
10	7, 9	sylibr 224	. . 3 |- ( A e. V -> { s e. (sigAlgebra ` U. A ) | A C_ s } e. ~P (sigAlgebra ` U. A ) )
11		insiga 30200	. . 3 |- ( ( { s e. (sigAlgebra ` U. A ) | A C_ s } =/= (/) /\ { s e. (sigAlgebra ` U. A ) | A C_ s } e. ~P (sigAlgebra ` U. A ) ) -> |^| { s e. (sigAlgebra ` U. A ) | A C_ s } e. (sigAlgebra ` U. A ) )
12	5, 10, 11	syl2anc 693	. 2 |- ( A e. V -> |^| { s e. (sigAlgebra ` U. A ) | A C_ s } e. (sigAlgebra ` U. A ) )
13	1, 12	eqeltrd 2701	1 |- ( A e. V -> (sigaGen ` A ) e. (sigAlgebra ` U. A ) )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ` cfv 5888  sigAlgebracsiga 30170  sigaGencsigagen 30201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-siga 30171  df-sigagen 30202

This theorem is referenced by:  sgsiga  30205  unisg  30206  sigagenss2  30213  brsiga  30246  brsigarn  30247  cldssbrsiga  30250  sxsiga  30254  cnmbfm  30325  sxbrsiga  30352