Theorem dfdm3 4540

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 | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4373	. 2 |- dom A = { x | E. y x A y }
2		df-br 3786	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1536	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2194	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2101	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   <.cop 3401   class class class wbr 3785   dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-br 3786  df-dm 4373

This theorem is referenced by:  csbdmg  4547