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Theorem eldmressnsn 28089
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 3678 . 2  |-  ( A  e.  dom  F  ->  A  e.  { A } )
2 dmressnsn 28088 . 2  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
31, 2eleqtrrd 2373 1  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   {csn 3653   dom cdm 4705    |` cres 4707
This theorem is referenced by:  dfdfat2  28099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-res 4717
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