Theorem omedm 40713

Description: The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
omedm	|- ( O e. OutMeas -> dom O = ~P U. dom O )

Proof of Theorem omedm

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

Step	Hyp	Ref	Expression
1		isome 40708	. . . 4 |- ( O e. OutMeas -> ( O e. OutMeas <-> ( ( ( ( O : dom O --> ( 0 [,] +oo ) /\ dom O = ~P U. dom O ) /\ ( O ` (/) ) = 0 ) /\ A. x e. ~P U. dom O A. y e. ~P x ( O ` y ) <_ ( O ` x ) ) /\ A. x e. ~P dom O ( x ~<_ om -> ( O ` U. x ) <_ (Σ^ ` ( O |` x ) ) ) ) ) )
2	1	ibi 256	. . 3 |- ( O e. OutMeas -> ( ( ( ( O : dom O --> ( 0 [,] +oo ) /\ dom O = ~P U. dom O ) /\ ( O ` (/) ) = 0 ) /\ A. x e. ~P U. dom O A. y e. ~P x ( O ` y ) <_ ( O ` x ) ) /\ A. x e. ~P dom O ( x ~<_ om -> ( O ` U. x ) <_ (Σ^ ` ( O |` x ) ) ) ) )
3	2	simplld 791	. 2 |- ( O e. OutMeas -> ( ( O : dom O --> ( 0 [,] +oo ) /\ dom O = ~P U. dom O ) /\ ( O ` (/) ) = 0 ) )
4	3	simplrd 793	1 |- ( O e. OutMeas -> dom O = ~P U. dom O )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,]cicc 12178  Σ^{^}csumge0 40579  OutMeascome 40703

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-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-ome 40704

This theorem is referenced by:  caragenss  40718  omeunile  40719