Theorem dmsigagen 30207

Description: A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Assertion

Ref	Expression
dmsigagen	|- dom sigaGen = _V

Proof of Theorem dmsigagen

Dummy variables j s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vuniex 6954	. . . . . 6 |- U. j e. _V
2		pwsiga 30193	. . . . . 6 |- ( U. j e. _V -> ~P U. j e. (sigAlgebra ` U. j ) )
3	1, 2	ax-mp 5	. . . . 5 |- ~P U. j e. (sigAlgebra ` U. j )
4		pwuni 4474	. . . . 5 |- j C_ ~P U. j
5		sseq2 3627	. . . . . 6 |- ( s = ~P U. j -> ( j C_ s <-> j C_ ~P U. j ) )
6	5	rspcev 3309	. . . . 5 |- ( ( ~P U. j e. (sigAlgebra ` U. j ) /\ j C_ ~P U. j ) -> E. s e. (sigAlgebra ` U. j ) j C_ s )
7	3, 4, 6	mp2an 708	. . . 4 |- E. s e. (sigAlgebra ` U. j ) j C_ s
8		rabn0 3958	. . . 4 |- ( { s e. (sigAlgebra ` U. j ) | j C_ s } =/= (/) <-> E. s e. (sigAlgebra ` U. j ) j C_ s )
9	7, 8	mpbir 221	. . 3 |- { s e. (sigAlgebra ` U. j ) | j C_ s } =/= (/)
10		intex 4820	. . 3 |- ( { s e. (sigAlgebra ` U. j ) | j C_ s } =/= (/) <-> |^| { s e. (sigAlgebra ` U. j ) | j C_ s } e. _V )
11	9, 10	mpbi 220	. 2 |- |^| { s e. (sigAlgebra ` U. j ) | j C_ s } e. _V
12		df-sigagen 30202	. 2 |- sigaGen = ( j e. _V |-> |^| { s e. (sigAlgebra ` U. j ) | j C_ s } )
13	11, 12	dmmpti 6023	1 |- dom sigaGen = _V

Syntax hints:   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   dom cdm 5114   ` 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-fn 5891  df-fv 5896  df-siga 30171  df-sigagen 30202

This theorem is referenced by: (None)