MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdm3 Unicode version

Theorem dfdm3 4867
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 4699 . 2  |-  dom  A  =  { x  |  E. y  x A y }
2 df-br 4024 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1569 . . 3  |-  ( E. y  x A y  <->  E. y <. x ,  y
>.  e.  A )
43abbii 2395 . 2  |-  { x  |  E. y  x A y }  =  {
x  |  E. y <. x ,  y >.  e.  A }
51, 4eqtri 2303 1  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   <.cop 3643   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  prjdmn  25082  csbdmg  27980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-br 4024  df-dm 4699
  Copyright terms: Public domain W3C validator