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Theorem braew 29017
 Description: 'almost everywhere' relation for a measure and a property (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypothesis
Ref Expression
braew.1
Assertion
Ref Expression
braew measures a.e.
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem braew
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 braew.1 . . . . 5
2 dmexg 6682 . . . . . 6 measures
3 uniexg 6546 . . . . . 6
42, 3syl 17 . . . . 5 measures
51, 4syl5eqelr 2511 . . . 4 measures
6 rabexg 4517 . . . 4
75, 6syl 17 . . 3 measures
8 simpr 462 . . . . . 6
98dmeqd 4999 . . . . . . . 8
109unieqd 4172 . . . . . . 7
11 simpl 458 . . . . . . 7
1210, 11difeq12d 3527 . . . . . 6
138, 12fveq12d 5831 . . . . 5
1413eqeq1d 2430 . . . 4
15 df-ae 29014 . . . 4 a.e.
1614, 15brabga 4677 . . 3 measures a.e.
177, 16mpancom 673 . 2 measures a.e.
181difeq1i 3522 . . . . 5
19 notrab 3693 . . . . 5
2018, 19eqtri 2450 . . . 4
2120fveq2i 5828 . . 3
2221eqeq1i 2433 . 2
2317, 22syl6bb 264 1 measures a.e.
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   wceq 1437   wcel 1872  crab 2718  cvv 3022   cdif 3376  cuni 4162   class class class wbr 4366   cdm 4796   crn 4797  cfv 5544  cc0 9490  measurescmeas 28969  a.e.cae 29012 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-cnv 4804  df-dm 4806  df-rn 4807  df-iota 5508  df-fv 5552  df-ae 29014 This theorem is referenced by:  truae  29018  aean  29019
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