Theorem dfdm3 4849

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 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4669	. 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2		df-br 4030	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	exbii 1616	. . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
4	3	abbii 2309	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	1, 4	eqtri 2214	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}

Syntax hints:   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ⟨cop 3621   class class class wbr 4029  dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-br 4030  df-dm 4669

This theorem is referenced by:  csbdmg  4856