Theorem dfdm3 5310

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 5124	. 2 |- dom A = { x | E. y x A y }
2		df-br 4654	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1774	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2739	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2644	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   <.cop 4183   class class class wbr 4653   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-br 4654  df-dm 5124

This theorem is referenced by:  csbdm  5318  cnextf  21870