MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldmg Structured version   Visualization version   GIF version

Theorem eldmg 5898
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldmg (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldmg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq1 5151 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝑦𝐴𝐵𝑦))
21exbidv 1924 . 2 (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3 df-dm 5686 . 2 dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
42, 3elab2g 3670 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wex 1781  wcel 2106   class class class wbr 5148  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-dm 5686
This theorem is referenced by:  eldm2g  5899  eldm  5900  breldmg  5909  releldmb  5945  funeu  6573  fneu  6659  ndmfv  6926  erref  8722  ecdmn0  8749  rlimdm  15494  rlimdmo1  15561  iscmet3lem2  24808  dvcnp2  25436  ulmcau  25906  pserulm  25933  mulog2sum  27037  gg-dvcnp2  35169  unbdqndv1  35379  eldmres  37133  eldmressnALTV  37135  eldm4  37137  eldmres2  37138  eldmcnv  37209  funressneu  45747  afveu  45851  rlimdmafv  45875  funressndmafv2rn  45921  afv2eu  45936  rlimdmafv2  45956
  Copyright terms: Public domain W3C validator