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Theorem dfdm3 4549
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 4382 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2 df-br 3792 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1512 . . 3 (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2169 . 2 {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2076 1 dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wex 1397  wcel 1409  {cab 2042  cop 3405   class class class wbr 3791  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-br 3792  df-dm 4382
This theorem is referenced by:  csbdmg  4556
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