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Theorem dfdm3 5060
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 4890 . 2  |-  dom  A  =  { x  |  E. y  x A y }
2 df-br 4215 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1593 . . 3  |-  ( E. y  x A y  <->  E. y <. x ,  y
>.  e.  A )
43abbii 2550 . 2  |-  { x  |  E. y  x A y }  =  {
x  |  E. y <. x ,  y >.  e.  A }
51, 4eqtri 2458 1  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   <.cop 3819   class class class wbr 4214   dom cdm 4880
This theorem is referenced by:  cnextf  18099  csbdmg  27960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-br 4215  df-dm 4890
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