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Theorem eldm2g 4873
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 4872 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2 df-br 4044 . . 3 (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
32exbii 1627 . 2 (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
41, 3bitrdi 196 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wex 1514  wcel 2175  cop 3635   class class class wbr 4043  dom cdm 4674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-dm 4684
This theorem is referenced by:  eldm2  4875  opeldmg  4882  dmfco  5646  releldm2  6270  tfrlem9  6404  climcau  11629  lmff  14692
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