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Theorem eldmg 5308
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 4647 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝑦𝐴𝐵𝑦))
21exbidv 1848 . 2 (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3 df-dm 5114 . 2 dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
42, 3elab2g 3347 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wex 1702  wcel 1988   class class class wbr 4644  dom cdm 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-dm 5114
This theorem is referenced by:  eldm2g  5309  eldm  5310  breldmg  5319  releldmb  5349  funeu  5901  fneu  5983  ndmfv  6205  erref  7747  ecdmn0  7774  rlimdm  14263  rlimdmo1  14329  iscmet3lem2  23071  dvcnp2  23664  ulmcau  24130  pserulm  24157  mulog2sum  25207  unbdqndv1  32474  afveu  40996  rlimdmafv  41020
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