Theorem dfdm3 2522

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.

Assertion

Ref	Expression
dfdm3	⊢ dom A = {x∣∃y⟨x, y⟩ ∈ A}

Distinct variable group(s):   x,y,A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 2428	. 2 ⊢ dom A = {x∣∃y xAy}
2		df-br 2063	. . . 4 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
3	2	biex 733	. . 3 ⊢ (∃y xAy ↔ ∃y⟨x, y⟩ ∈ A)
4	3	biabi 1181	. 2 ⊢ {x∣∃y xAy} = {x∣∃y⟨x, y⟩ ∈ A}
5	1, 4	eqtr 1119	1 ⊢ dom A = {x∣∃y⟨x, y⟩ ∈ A}

Syntax hints:  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  dom cdm 2410

This theorem is referenced by:  dfdmf 2526  dm0 2542  dmsn0 2543  dmsnsn0 2544  dmsnop 2547  dmexg 2551

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-br 2063  df-dm 2428

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