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Theorem eldmeldmressn 5320
Description: An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
eldmeldmressn  |-  ( X  e.  dom  F  <->  X  e.  dom  ( F  |`  { X } ) )

Proof of Theorem eldmeldmressn
StepHypRef Expression
1 eldmressnsn 5319 . 2  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  |`  { X } ) )
2 elin 3692 . . . 4  |-  ( X  e.  ( { X }  i^i  dom  F )  <->  ( X  e.  { X }  /\  X  e.  dom  F ) )
32simprbi 464 . . 3  |-  ( X  e.  ( { X }  i^i  dom  F )  ->  X  e.  dom  F
)
4 dmres 5300 . . 3  |-  dom  ( F  |`  { X }
)  =  ( { X }  i^i  dom  F )
53, 4eleq2s 2575 . 2  |-  ( X  e.  dom  ( F  |`  { X } )  ->  X  e.  dom  F )
61, 5impbii 188 1  |-  ( X  e.  dom  F  <->  X  e.  dom  ( F  |`  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1767    i^i cin 3480   {csn 4033   dom cdm 5005    |` cres 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-dm 5015  df-res 5017
This theorem is referenced by:  fvn0fvelrn  6089
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