Theorem measbase 30260

Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
measbase	|- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )

Proof of Theorem measbase

Dummy variables x m y s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. 2 |- ( M e. (measures ` S ) -> S e. dom measures )
2		vex 3203	. . . . 5 |- s e. _V
3		ovex 6678	. . . . 5 |- ( 0 [,] +oo ) e. _V
4		mapex 7863	. . . . 5 |- ( ( s e. _V /\ ( 0 [,] +oo ) e. _V ) -> { m | m : s --> ( 0 [,] +oo ) } e. _V )
5	2, 3, 4	mp2an 708	. . . 4 |- { m | m : s --> ( 0 [,] +oo ) } e. _V
6		simp1 1061	. . . . 5 |- ( ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) -> m : s --> ( 0 [,] +oo ) )
7	6	ss2abi 3674	. . . 4 |- { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } C_ { m | m : s --> ( 0 [,] +oo ) }
8	5, 7	ssexi 4803	. . 3 |- { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } e. _V
9		df-meas 30259	. . 3 |- measures = ( s e. U. ran sigAlgebra |-> { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } )
10	8, 9	dmmpti 6023	. 2 |- dom measures = U. ran sigAlgebra
11	1, 10	syl6eleq 2711	1 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  Σ^*cesum 30089  sigAlgebracsiga 30170  measurescmeas 30258

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-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-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-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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-meas 30259

This theorem is referenced by:  measfrge0  30266  measvnul  30269  measvun  30272  measxun2  30273  measun  30274  measvuni  30277  measssd  30278  measunl  30279  measiuns  30280  measiun  30281  meascnbl  30282  measinblem  30283  measinb  30284  measinb2  30286  measdivcstOLD  30287  measdivcst  30288  aean  30307  mbfmbfm  30320  domprobsiga  30473  prob01  30475  probfinmeasbOLD  30490  probfinmeasb  30491  probmeasb  30492