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Theorem dmressnsn 5300
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 5282 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
2 snssi 4160 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
3 df-ss 3475 . . 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 2507 1  |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   {csn 4016   dom cdm 4988    |` cres 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998  df-res 5000
This theorem is referenced by:  eldmressnsn  5301  funcoressn  32454  funressnfv  32455
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