Theorem dfdm3 5557

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 5367	. 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
2		df-br 4889	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	exbii 1892	. . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
4	3	abbii 2908	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	1, 4	eqtri 2802	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}

Syntax hints:   = wceq 1601  ∃wex 1823   ∈ wcel 2107  {cab 2763  ⟨cop 4404   class class class wbr 4888  dom cdm 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-sb 2012  df-clab 2764  df-cleq 2770  df-br 4889  df-dm 5367

This theorem is referenced by:  csbdm  5565  cnextf  22289