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Theorem dfdm3 5878
Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfdm3 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfdm3
StepHypRef Expression
1 df-dm 5677 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2 df-br 5140 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1842 . . 3 (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2794 . 2 {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2752 1 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wex 1773  wcel 2098  {cab 2701  cop 4627   class class class wbr 5139  dom cdm 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-br 5140  df-dm 5677
This theorem is referenced by:  csbdm  5888  cnextf  23914  dmrab  32232
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