MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldmeldmressn Structured version   Unicode version

Theorem eldmeldmressn 5165
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 5164 . 2  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  |`  { X } ) )
2 elin 3655 . . . 4  |-  ( X  e.  ( { X }  i^i  dom  F )  <->  ( X  e.  { X }  /\  X  e.  dom  F ) )
32simprbi 465 . . 3  |-  ( X  e.  ( { X }  i^i  dom  F )  ->  X  e.  dom  F
)
4 dmres 5145 . . 3  |-  dom  ( F  |`  { X }
)  =  ( { X }  i^i  dom  F )
53, 4eleq2s 2537 . 2  |-  ( X  e.  dom  ( F  |`  { X } )  ->  X  e.  dom  F )
61, 5impbii 190 1  |-  ( X  e.  dom  F  <->  X  e.  dom  ( F  |`  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1870    i^i cin 3441   {csn 4002   dom cdm 4854    |` cres 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-dm 4864  df-res 4866
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator