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Theorem eldmressn 28087
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 3371 . . 3  |-  ( B  e.  ( { A }  i^i  dom  F )  <->  ( B  e.  { A }  /\  B  e.  dom  F ) )
2 elsni 3677 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
32adantr 451 . . 3  |-  ( ( B  e.  { A }  /\  B  e.  dom  F )  ->  B  =  A )
41, 3sylbi 187 . 2  |-  ( B  e.  ( { A }  i^i  dom  F )  ->  B  =  A )
5 dmres 4992 . 2  |-  dom  ( F  |`  { A }
)  =  ( { A }  i^i  dom  F )
64, 5eleq2s 2388 1  |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164   {csn 3653   dom cdm 4705    |` cres 4707
This theorem is referenced by:  dfdfat2  28099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-res 4717
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