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Theorem measval 29094
 Description: The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
Assertion
Ref Expression
measval sigAlgebra measures Disj Σ*
Distinct variable groups:   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem measval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simp1 1030 . . . 4 Disj Σ*
21ss2abi 3487 . . 3 Disj Σ*
3 ovex 6336 . . . 4
4 mapex 7496 . . . 4 sigAlgebra
53, 4mpan2 685 . . 3 sigAlgebra
6 ssexg 4542 . . 3 Disj Σ* Disj Σ*
72, 5, 6sylancr 676 . 2 sigAlgebra Disj Σ*
8 feq2 5721 . . . . 5
9 pweq 3945 . . . . . 6
109raleqdv 2979 . . . . 5 Disj Σ* Disj Σ*
118, 103anbi13d 1367 . . . 4 Disj Σ* Disj Σ*
1211abbidv 2589 . . 3 Disj Σ* Disj Σ*
13 df-meas 29092 . . 3 measures sigAlgebra Disj Σ*
1412, 13fvmptg 5961 . 2 sigAlgebra Disj Σ* measures Disj Σ*
157, 14mpdan 681 1 sigAlgebra measures Disj Σ*
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   w3a 1007   wceq 1452   wcel 1904  cab 2457  wral 2756  cvv 3031   wss 3390  c0 3722  cpw 3942  cuni 4190  Disj wdisj 4366   class class class wbr 4395   crn 4840  wf 5585  cfv 5589  (class class class)co 6308  com 6711   cdom 7585  cc0 9557   cpnf 9690  cicc 11663  Σ*cesum 28922  sigAlgebracsiga 29003  measurescmeas 29091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-meas 29092 This theorem is referenced by: (None)
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