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Theorem eldmressnsn 27962
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 3839 . 2  |-  ( A  e.  dom  F  ->  A  e.  { A } )
2 dmressnsn 27961 . 2  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
31, 2eleqtrrd 2513 1  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   {csn 3814   dom cdm 4878    |` cres 4880
This theorem is referenced by:  dfdfat2  27971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-res 4890
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