Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  measbasedom Structured version   Unicode version

Theorem measbasedom 26752
 Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom measures measures

Proof of Theorem measbasedom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 26750 . . . 4 measures sigAlgebra Disj Σ*
21simprd 463 . . 3 measures Disj Σ*
31simpld 459 . . . 4 measures sigAlgebra
4 ismeas 26749 . . . 4 sigAlgebra measures Disj Σ*
53, 4syl 16 . . 3 measures measures Disj Σ*
62, 5mpbird 232 . 2 measures measures
7 df-meas 26746 . . . 4 measures sigAlgebra Disj Σ*
87funmpt2 5555 . . 3 measures
9 elunirn2 26102 . . 3 measures measures measures
108, 9mpan 670 . 2 measures measures
116, 10impbii 188 1 measures measures
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 965   wceq 1370   wcel 1758  cab 2436  wral 2795  c0 3737  cpw 3960  cuni 4191  Disj wdisj 4362   class class class wbr 4392   cdm 4940   crn 4941   wfun 5512  wf 5514  cfv 5518  (class class class)co 6192  com 6578   cdom 7410  cc0 9385   cpnf 9518  cicc 11406  Σ*cesum 26619  sigAlgebracsiga 26686  measurescmeas 26745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-esum 26620  df-meas 26746 This theorem is referenced by:  truae  26795  aean  26796  mbfmbfm  26809  sibfinima  26861  sibfof  26862  domprobmeas  26929  probmeasd  26942  probfinmeasbOLD  26947  probfinmeasb  26948  probmeasb  26949  dstrvprob  26990
 Copyright terms: Public domain W3C validator