Theorem dfdm3 4726

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 4549	. 2 |- dom A = { x | E. y x A y }
2		df-br 3930	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1584	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2255	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2160	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   <.cop 3530   class class class wbr 3929   dom cdm 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-br 3930  df-dm 4549

This theorem is referenced by:  csbdmg  4733