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Theorem dfdm3 5216
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 5034 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2 df-br 4574 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1762 . . 3 (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2721 . 2 {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2627 1 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  wcel 1975  {cab 2591  cop 4126   class class class wbr 4573  dom cdm 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-br 4574  df-dm 5034
This theorem is referenced by:  csbdm  5223  cnextf  21618
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