Theorem brae 32258

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

Assertion

Ref	Expression
brae	⊢ ((𝑀 ∈ ∪ ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘(∪ 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 486	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
2	1	dmeqd 5827	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
3	2	unieqd 4858	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
4		simpl 484	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → 𝑎 = 𝐴)
5	3, 4	difeq12d 4064	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → (∪ dom 𝑚 ∖ 𝑎) = (∪ dom 𝑀 ∖ 𝐴))
6	1, 5	fveq12d 6811	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = (𝑀‘(∪ dom 𝑀 ∖ 𝐴)))
7	6	eqeq1d 2738	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑀) → ((𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0 ↔ (𝑀‘(∪ dom 𝑀 ∖ 𝐴)) = 0))
8		df-ae 32256	. . 3 ⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
9	7, 8	brabga 5460	. 2 ⊢ ((𝐴 ∈ dom 𝑀 ∧ 𝑀 ∈ ∪ ran measures) → (𝐴a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ 𝐴)) = 0))
10	9	ancoms 460	1 ⊢ ((𝑀 ∈ ∪ ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ 𝐴)) = 0))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104   ∖ cdif 3889  ∪ cuni 4844   class class class wbr 5081  dom cdm 5600  ran crn 5601  ‘cfv 6458  0cc0 10921  measurescmeas 32212  a.e.cae 32254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-dm 5610  df-iota 6410  df-fv 6466  df-ae 32256

This theorem is referenced by: (None)