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Theorem prjdmn 25082
Description: The projection of the first elements of the pairs of a relation  R is its domain. (Contributed by FL, 5-Oct-2009.)
Assertion
Ref Expression
prjdmn  |-  ( Rel 
R  ->  ( 1st " R )  =  dom  R )

Proof of Theorem prjdmn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prj1b 25079 . 2  |-  ( Rel 
R  ->  ( 1st " R )  =  {
x  |  E. y <. x ,  y >.  e.  R } )
2 dfdm3 4867 . 2  |-  dom  R  =  { x  |  E. y <. x ,  y
>.  e.  R }
31, 2syl6eqr 2333 1  |-  ( Rel 
R  ->  ( 1st " R )  =  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   <.cop 3643   dom cdm 4689   "cima 4692   Rel wrel 4694   1stc1st 6120
This theorem is referenced by:  prjcp1  25084  relinccppr  25129
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123
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