Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  volres Structured version   Unicode version

Theorem volres 22233
 Description: A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
Assertion
Ref Expression
volres

Proof of Theorem volres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resdmres 5316 . 2
2 df-vol 22171 . . . 4
32dmeqi 5027 . . 3
43reseq2i 5093 . 2
51, 4, 23eqtr4ri 2444 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1407  cab 2389  wral 2756   cdif 3413   cin 3415  ccnv 4824   cdm 4825   cres 4827  cima 4828  cfv 5571  (class class class)co 6280  cr 9523   caddc 9527  covol 22168  cvol 22169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-vol 22171 This theorem is referenced by:  volf  22234  mblvol  22235
 Copyright terms: Public domain W3C validator