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Theorem dfdm3 4869
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
StepHypRef Expression
1 df-dm 4701 . 2  |-  dom  A  =  { x  |  E. y  x A y }
2 df-br 4026 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1571 . . 3  |-  ( E. y  x A y  <->  E. y <. x ,  y
>.  e.  A )
43abbii 2397 . 2  |-  { x  |  E. y  x A y }  =  {
x  |  E. y <. x ,  y >.  e.  A }
51, 4eqtri 2305 1  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:   E.wex 1530    = wceq 1625    e. wcel 1686   {cab 2271   <.cop 3645   class class class wbr 4025   dom cdm 4691
This theorem is referenced by:  prjdmn  25093  csbdmg  27991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-br 4026  df-dm 4701
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