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Theorem eldm2g 4531
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 4530 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2 df-br 3765 . . 3 (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
32exbii 1496 . 2 (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
41, 3syl6bb 185 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wex 1381  wcel 1393  cop 3378   class class class wbr 3764  dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355
This theorem is referenced by:  eldm2  4533  opeldmg  4540  dmfco  5241  releldm2  5811  tfrlem9  5935  climcau  9840
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