Theorem dfdm3 4465

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 4298	. 2 ⊢ dom A = {x ∣ ∃y xAy}
2		df-br 3756	. . . 4 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
3	2	exbii 1493	. . 3 ⊢ (∃y xAy ↔ ∃y⟨x, y⟩ ∈ A)
4	3	abbii 2150	. 2 ⊢ {x ∣ ∃y xAy} = {x ∣ ∃y⟨x, y⟩ ∈ A}
5	1, 4	eqtri 2057	1 ⊢ dom A = {x ∣ ∃y⟨x, y⟩ ∈ A}

Syntax hints:   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ⟨cop 3370   class class class wbr 3755  dom cdm 4288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-br 3756  df-dm 4298

This theorem is referenced by:  csbdmg  4472