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Theorem dfdm3 5746
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 5553 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2 df-br 5054 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1849 . . 3 (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2889 . 2 {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2847 1 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wex 1781  wcel 2115  {cab 2802  cop 4556   class class class wbr 5053  dom cdm 5543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-br 5054  df-dm 5553
This theorem is referenced by:  csbdm  5754  cnextf  22669  dmrab  30264
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