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Theorem eldmg 4734
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 3932 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝑦𝐴𝐵𝑦))
21exbidv 1797 . 2 (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3 df-dm 4549 . 2 dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
42, 3elab2g 2831 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wex 1468  wcel 1480   class class class wbr 3929  dom cdm 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549
This theorem is referenced by:  eldm2g  4735  eldm  4736  breldmg  4745  releldmb  4776  funeu  5148  fneu  5227  ndmfvg  5452  erref  6449  ecdmn0m  6471  shftdm  10606  dvcnp2cntop  12846
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