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Theorem dfdm3 4522
 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 4355 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2 df-br 3765 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1496 . . 3 (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2153 . 2 {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2060 1 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ⟨cop 3378   class class class wbr 3764  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-br 3765  df-dm 4355 This theorem is referenced by:  csbdmg  4529
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