Theorem brae 30304

Description: 'almost everywhere' relation for a measure and a measurable set A. (Contributed by Thierry Arnoux, 20-Oct-2017.)

Assertion

Ref	Expression
brae	|- ( ( M e. U. ran measures /\ A e. dom M ) -> ( Aa.e. M <-> ( M ` ( U. dom M \ A ) ) = 0 ) )

Proof of Theorem brae

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

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( a = A /\ m = M ) -> m = M )
2	1	dmeqd 5326	. . . . . . 7 |- ( ( a = A /\ m = M ) -> dom m = dom M )
3	2	unieqd 4446	. . . . . 6 |- ( ( a = A /\ m = M ) -> U. dom m = U. dom M )
4		simpl 473	. . . . . 6 |- ( ( a = A /\ m = M ) -> a = A )
5	3, 4	difeq12d 3729	. . . . 5 |- ( ( a = A /\ m = M ) -> ( U. dom m \ a ) = ( U. dom M \ A ) )
6	1, 5	fveq12d 6197	. . . 4 |- ( ( a = A /\ m = M ) -> ( m ` ( U. dom m \ a ) ) = ( M ` ( U. dom M \ A ) ) )
7	6	eqeq1d 2624	. . 3 |- ( ( a = A /\ m = M ) -> ( ( m ` ( U. dom m \ a ) ) = 0 <-> ( M ` ( U. dom M \ A ) ) = 0 ) )
8		df-ae 30302	. . 3 |- a.e. = { <. a , m >. | ( m ` ( U. dom m \ a ) ) = 0 }
9	7, 8	brabga 4989	. 2 |- ( ( A e. dom M /\ M e. U. ran measures ) -> ( Aa.e. M <-> ( M ` ( U. dom M \ A ) ) = 0 ) )
10	9	ancoms 469	1 |- ( ( M e. U. ran measures /\ A e. dom M ) -> ( Aa.e. M <-> ( M ` ( U. dom M \ A ) ) = 0 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ 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-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

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-dm 5124  df-iota 5851  df-fv 5896  df-ae 30302

This theorem is referenced by: (None)