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Theorem eldmressn 31998
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 3692 . . 3  |-  ( B  e.  ( { A }  i^i  dom  F )  <->  ( B  e.  { A }  /\  B  e.  dom  F ) )
2 elsni 4058 . . . 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 5300 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
64, 5eleq2s 2575 1  |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480   {csn 4033   dom cdm 5005    |` cres 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-dm 5015  df-res 5017
This theorem is referenced by:  dfdfat2  32006
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