Theorem dfdm3 5859

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 5653	. 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2		df-br 5098	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	exbii 1867	. . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
4	3	abbii 2828	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	1, 4	eqtri 2784	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}

Syntax hints:   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ⟨cop 4585   class class class wbr 5097  dom cdm 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-br 5098  df-dm 5653

This theorem is referenced by:  csbdm  5869  cnextf  24114  dmrab  32655