Theorem truae 30306

Description: A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)

Hypotheses

Ref	Expression
truae.1	|- U. dom M = O
truae.2	|- ( ph -> M e. U. ran measures )
truae.3	|- ( ph -> ps )

Assertion

Ref	Expression
truae	|- ( ph -> { x e. O | ps }a.e. M )

Distinct variable groups:   x, O   ph, x

Allowed substitution hints:   ps( x)   M( x)

Proof of Theorem truae

Step	Hyp	Ref	Expression
1		truae.3	. . . . . . . 8 |- ( ph -> ps )
2	1	pm2.24d 147	. . . . . . 7 |- ( ph -> ( -. ps -> x e. (/) ) )
3	2	ralrimivw 2967	. . . . . 6 |- ( ph -> A. x e. O ( -. ps -> x e. (/) ) )
4		rabss 3679	. . . . . 6 |- ( { x e. O | -. ps } C_ (/) <-> A. x e. O ( -. ps -> x e. (/) ) )
5	3, 4	sylibr 224	. . . . 5 |- ( ph -> { x e. O | -. ps } C_ (/) )
6		ss0 3974	. . . . 5 |- ( { x e. O | -. ps } C_ (/) -> { x e. O | -. ps } = (/) )
7	5, 6	syl 17	. . . 4 |- ( ph -> { x e. O | -. ps } = (/) )
8	7	fveq2d 6195	. . 3 |- ( ph -> ( M ` { x e. O | -. ps } ) = ( M ` (/) ) )
9		truae.2	. . . 4 |- ( ph -> M e. U. ran measures )
10		measbasedom 30265	. . . . 5 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
11		measvnul 30269	. . . . 5 |- ( M e. (measures ` dom M ) -> ( M ` (/) ) = 0 )
12	10, 11	sylbi 207	. . . 4 |- ( M e. U. ran measures -> ( M ` (/) ) = 0 )
13	9, 12	syl 17	. . 3 |- ( ph -> ( M ` (/) ) = 0 )
14	8, 13	eqtrd 2656	. 2 |- ( ph -> ( M ` { x e. O | -. ps } ) = 0 )
15		truae.1	. . . 4 |- U. dom M = O
16	15	braew 30305	. . 3 |- ( M e. U. ran measures -> ( { x e. O | ps }a.e. M <-> ( M ` { x e. O | -. ps } ) = 0 ) )
17	9, 16	syl 17	. 2 |- ( ph -> ( { x e. O | ps }a.e. M <-> ( M ` { x e. O | -. ps } ) = 0 ) )
18	14, 17	mpbird 247	1 |- ( ph -> { x e. O | ps }a.e. M )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   ` cfv 5888   0cc0 9936  measurescmeas 30258  a.e.cae 30300

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-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-esum 30090  df-meas 30259  df-ae 30302

This theorem is referenced by: (None)