Theorem caragensspw 40723

Description: The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caragensspw.o	|- ( ph -> O e. OutMeas )
caragensspw.x	|- X = U. dom O
caragensspw.s	|- S = (CaraGen ` O )

Assertion

Ref	Expression
caragensspw	|- ( ph -> S C_ ~P X )

Proof of Theorem caragensspw

Step	Hyp	Ref	Expression
1		caragensspw.o	. . . 4 |- ( ph -> O e. OutMeas )
2		caragensspw.s	. . . . 5 |- S = (CaraGen ` O )
3	2	caragenss 40718	. . . 4 |- ( O e. OutMeas -> S C_ dom O )
4	1, 3	syl 17	. . 3 |- ( ph -> S C_ dom O )
5		pwuni 4474	. . . 4 |- dom O C_ ~P U. dom O
6	5	a1i 11	. . 3 |- ( ph -> dom O C_ ~P U. dom O )
7	4, 6	sstrd 3613	. 2 |- ( ph -> S C_ ~P U. dom O )
8		caragensspw.x	. . . . 5 |- X = U. dom O
9	8	pweqi 4162	. . . 4 |- ~P X = ~P U. dom O
10	9	eqcomi 2631	. . 3 |- ~P U. dom O = ~P X
11	10	a1i 11	. 2 |- ( ph -> ~P U. dom O = ~P X )
12	7, 11	sseqtrd 3641	1 |- ( ph -> S C_ ~P X )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   ` cfv 5888  OutMeascome 40703  CaraGenccaragen 40705

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-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-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-ome 40704  df-caragen 40706

This theorem is referenced by:  caratheodorylem2  40741