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Theorem eldmressnsn 27984
Description: The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
eldmressnsn  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )

Proof of Theorem eldmressnsn
StepHypRef Expression
1 snidg 3665 . 2  |-  ( A  e.  dom  F  ->  A  e.  { A } )
2 dmressnsn 27983 . 2  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
31, 2eleqtrrd 2360 1  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   {csn 3640   dom cdm 4689    |` cres 4691
This theorem is referenced by:  dfdfat2  27994
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-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-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-res 4701
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