Theorem braew 30305

Description: 'almost everywhere' relation for a measure M and a property ph (Contributed by Thierry Arnoux, 20-Oct-2017.)

Hypothesis

Ref	Expression
braew.1	|- U. dom M = O

Assertion

Ref	Expression
braew	|- ( M e. U. ran measures -> ( { x e. O | ph }a.e. M <-> ( M ` { x e. O | -. ph } ) = 0 ) )

Distinct variable group:   x, O

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

Proof of Theorem braew

Dummy variables m a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		braew.1	. . . . 5 |- U. dom M = O
2		dmexg 7097	. . . . . 6 |- ( M e. U. ran measures -> dom M e. _V )
3		uniexg 6955	. . . . . 6 |- ( dom M e. _V -> U. dom M e. _V )
4	2, 3	syl 17	. . . . 5 |- ( M e. U. ran measures -> U. dom M e. _V )
5	1, 4	syl5eqelr 2706	. . . 4 |- ( M e. U. ran measures -> O e. _V )
6		rabexg 4812	. . . 4 |- ( O e. _V -> { x e. O | ph } e. _V )
7	5, 6	syl 17	. . 3 |- ( M e. U. ran measures -> { x e. O | ph } e. _V )
8		simpr 477	. . . . . 6 |- ( ( a = { x e. O | ph } /\ m = M ) -> m = M )
9	8	dmeqd 5326	. . . . . . . 8 |- ( ( a = { x e. O | ph } /\ m = M ) -> dom m = dom M )
10	9	unieqd 4446	. . . . . . 7 |- ( ( a = { x e. O | ph } /\ m = M ) -> U. dom m = U. dom M )
11		simpl 473	. . . . . . 7 |- ( ( a = { x e. O | ph } /\ m = M ) -> a = { x e. O | ph } )
12	10, 11	difeq12d 3729	. . . . . 6 |- ( ( a = { x e. O | ph } /\ m = M ) -> ( U. dom m \ a ) = ( U. dom M \ { x e. O | ph } ) )
13	8, 12	fveq12d 6197	. . . . 5 |- ( ( a = { x e. O | ph } /\ m = M ) -> ( m ` ( U. dom m \ a ) ) = ( M ` ( U. dom M \ { x e. O | ph } ) ) )
14	13	eqeq1d 2624	. . . 4 |- ( ( a = { x e. O | ph } /\ m = M ) -> ( ( m ` ( U. dom m \ a ) ) = 0 <-> ( M ` ( U. dom M \ { x e. O | ph } ) ) = 0 ) )
15		df-ae 30302	. . . 4 |- a.e. = { <. a , m >. | ( m ` ( U. dom m \ a ) ) = 0 }
16	14, 15	brabga 4989	. . 3 |- ( ( { x e. O | ph } e. _V /\ M e. U. ran measures ) -> ( { x e. O | ph }a.e. M <-> ( M ` ( U. dom M \ { x e. O | ph } ) ) = 0 ) )
17	7, 16	mpancom 703	. 2 |- ( M e. U. ran measures -> ( { x e. O | ph }a.e. M <-> ( M ` ( U. dom M \ { x e. O | ph } ) ) = 0 ) )
18	1	difeq1i 3724	. . . . 5 |- ( U. dom M \ { x e. O | ph } ) = ( O \ { x e. O | ph } )
19		notrab 3904	. . . . 5 |- ( O \ { x e. O | ph } ) = { x e. O | -. ph }
20	18, 19	eqtri 2644	. . . 4 |- ( U. dom M \ { x e. O | ph } ) = { x e. O | -. ph }
21	20	fveq2i 6194	. . 3 |- ( M ` ( U. dom M \ { x e. O | ph } ) ) = ( M ` { x e. O | -. ph } )
22	21	eqeq1i 2627	. 2 |- ( ( M ` ( U. dom M \ { x e. O | ph } ) ) = 0 <-> ( M ` { x e. O | -. ph } ) = 0 )
23	17, 22	syl6bb 276	1 |- ( M e. U. ran measures -> ( { x e. O | ph }a.e. M <-> ( M ` { x e. O | -. ph } ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ae 30302

This theorem is referenced by:  truae  30306  aean  30307