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Theorem eldmressn 30027
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 3539 . . 3  |-  ( B  e.  ( { A }  i^i  dom  F )  <->  ( B  e.  { A }  /\  B  e.  dom  F ) )
2 elsni 3902 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
32adantr 465 . . 3  |-  ( ( B  e.  { A }  /\  B  e.  dom  F )  ->  B  =  A )
41, 3sylbi 195 . 2  |-  ( B  e.  ( { A }  i^i  dom  F )  ->  B  =  A )
5 dmres 5131 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
64, 5eleq2s 2535 1  |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3327   {csn 3877   dom cdm 4840    |` cres 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-dm 4850  df-res 4852
This theorem is referenced by:  dfdfat2  30037
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