Theorem dfdm3 4522

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 4355	. 2 |- dom A = { x | E. y x A y }
2		df-br 3765	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1496	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2153	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2060	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   <.cop 3378   class class class wbr 3764   dom cdm 4345

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-br 3765  df-dm 4355

This theorem is referenced by:  csbdmg  4529