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Theorem elreimasng 4767
Description: Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
Assertion
Ref Expression
elreimasng  |-  ( ( Rel  R  /\  A  e.  V )  ->  ( B  e.  ( R " { A } )  <-> 
A R B ) )

Proof of Theorem elreimasng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imasng 4766 . . 3  |-  ( A  e.  V  ->  ( R " { A }
)  =  { x  |  A R x }
)
21eleq2d 2154 . 2  |-  ( A  e.  V  ->  ( B  e.  ( R " { A } )  <-> 
B  e.  { x  |  A R x }
) )
3 brrelex2 4451 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 113 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 3826 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 2757 . . 3  |-  ( ( A R B  ->  B  e.  _V )  ->  ( B  e.  {
x  |  A R x }  <->  A R B ) )
74, 6syl 14 . 2  |-  ( Rel 
R  ->  ( B  e.  { x  |  A R x }  <->  A R B ) )
82, 7sylan9bbr 451 1  |-  ( ( Rel  R  /\  A  e.  V )  ->  ( B  e.  ( R " { A } )  <-> 
A R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1436   {cab 2071   _Vcvv 2615   {csn 3431   class class class wbr 3822   "cima 4416   Rel wrel 4418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by: (None)
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