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Theorem dmressnsn 5105
Description: The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
dmressnsn  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )

Proof of Theorem dmressnsn
StepHypRef Expression
1 dmres 5087 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
2 snssi 4087 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
3 df-ss 3393 . . 3  |-  ( { A }  C_  dom  F  <-> 
( { A }  i^i  dom  F )  =  { A } )
42, 3sylib 199 . 2  |-  ( A  e.  dom  F  -> 
( { A }  i^i  dom  F )  =  { A } )
51, 4syl5eq 2474 1  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    i^i cin 3378    C_ wss 3379   {csn 3941   dom cdm 4796    |` cres 4798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-dm 4806  df-res 4808
This theorem is referenced by:  eldmressnsn  5106  funcoressn  38442  funressnfv  38443
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