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Theorem dmressnsn 30166
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 5229 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
2 snssi 4115 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
3 df-ss 3440 . . 3  |-  ( { A }  C_  dom  F  <-> 
( { A }  i^i  dom  F )  =  { A } )
42, 3sylib 196 . 2  |-  ( A  e.  dom  F  -> 
( { A }  i^i  dom  F )  =  { A } )
51, 4syl5eq 2504 1  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    i^i cin 3425    C_ wss 3426   {csn 3975   dom cdm 4938    |` cres 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-xp 4944  df-dm 4948  df-res 4950
This theorem is referenced by:  eldmressnsn  30167  funcoressn  30171  funressnfv  30172
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