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Theorem eldm 4824
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1 𝐴 ∈ V
Assertion
Ref Expression
eldm (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2 𝐴 ∈ V
2 eldmg 4822 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
31, 2ax-mp 5 1 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Colors of variables: wff set class
Syntax hints:  wb 105  wex 1492  wcel 2148  Vcvv 2737   class class class wbr 4003  dom cdm 4626
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-dm 4636
This theorem is referenced by:  dmi  4842  dmcoss  4896  dmcosseq  4898  dminss  5043  dmsnm  5094  dffun7  5243  dffun8  5244  fnres  5332  fndmdif  5621  reldmtpos  6253  dmtpos  6256  tfrexlem  6334
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