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Theorem eldmressn 27962
Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
eldmressn  |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )

Proof of Theorem eldmressn
StepHypRef Expression
1 elin 3532 . . 3  |-  ( B  e.  ( { A }  i^i  dom  F )  <->  ( B  e.  { A }  /\  B  e.  dom  F ) )
2 elsni 3840 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
32adantr 453 . . 3  |-  ( ( B  e.  { A }  /\  B  e.  dom  F )  ->  B  =  A )
41, 3sylbi 189 . 2  |-  ( B  e.  ( { A }  i^i  dom  F )  ->  B  =  A )
5 dmres 5169 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
64, 5eleq2s 2530 1  |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321   {csn 3816   dom cdm 4880    |` cres 4882
This theorem is referenced by:  dfdfat2  27973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-dm 4890  df-res 4892
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