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Theorem dfdm3 4550
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 4383 . 2  |-  dom  A  =  { x  |  E. y  x A y }
2 df-br 3793 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1512 . . 3  |-  ( E. y  x A y  <->  E. y <. x ,  y
>.  e.  A )
43abbii 2169 . 2  |-  { x  |  E. y  x A y }  =  {
x  |  E. y <. x ,  y >.  e.  A }
51, 4eqtri 2076 1  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   <.cop 3406   class class class wbr 3792   dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-br 3793  df-dm 4383
This theorem is referenced by:  csbdmg  4557
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