Theorem omexrcl 40721

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

Hypotheses

Ref	Expression
omexrcl.o	|- ( ph -> O e. OutMeas )
omexrcl.x	|- X = U. dom O
omexrcl.a	|- ( ph -> A C_ X )

Assertion

Ref	Expression
omexrcl	|- ( ph -> ( O ` A ) e. RR* )

Proof of Theorem omexrcl

Step	Hyp	Ref	Expression
1		iccssxr 12256	. 2 |- ( 0 [,] +oo ) C_ RR*
2		omexrcl.o	. . 3 |- ( ph -> O e. OutMeas )
3		omexrcl.x	. . 3 |- X = U. dom O
4		omexrcl.a	. . 3 |- ( ph -> A C_ X )
5	2, 3, 4	omecl 40717	. 2 |- ( ph -> ( O ` A ) e. ( 0 [,] +oo ) )
6	1, 5	sseldi 3601	1 |- ( ph -> ( O ` A ) e. RR* )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   dom cdm 5114   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   RR*cxr 10073   [,]cicc 12178  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-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  ax-cnex 9992  ax-resscn 9993

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-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-iun 4522  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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-xr 10078  df-icc 12182  df-ome 40704

This theorem is referenced by:  omessre  40724  caragenuncllem  40726  omeiunltfirp  40733  caratheodorylem1  40740  caratheodorylem2  40741  caragenel2d  40746  omess0  40748  caragencmpl  40749