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Theorem eldmg 5228
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 4580 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝑦𝐴𝐵𝑦))
21exbidv 1836 . 2 (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3 df-dm 5038 . 2 dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
42, 3elab2g 3321 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wex 1694  wcel 1976   class class class wbr 4577  dom cdm 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-dm 5038
This theorem is referenced by:  eldm2g  5229  eldm  5230  breldmg  5239  releldmb  5268  funeu  5814  fneu  5895  ndmfv  6113  erref  7627  ecdmn0  7654  rlimdm  14079  rlimdmo1  14145  iscmet3lem2  22843  dvcnp2  23434  ulmcau  23898  pserulm  23925  mulog2sum  24971  unbdqndv1  31503  afveu  39707  rlimdmafv  39731
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