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Theorem eldm2g 5910
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldm2g (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldm2g
StepHypRef Expression
1 eldmg 5909 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2 df-br 5144 . . 3 (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
32exbii 1848 . 2 (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
41, 3bitrdi 287 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1779  wcel 2108  cop 4632   class class class wbr 5143  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-dm 5695
This theorem is referenced by:  eldm2  5912  opeldmd  5917  dmfco  7005  releldm2  8068  tfrlem9  8425  climcau  15707  caucvgb  15716  lmff  23309  axhcompl-zf  31017  satfdmlem  35373  dfatdmfcoafv2  47266
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