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Theorem elreimasng 4715
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 4714 . . 3  |-  ( A  e.  V  ->  ( R " { A }
)  =  { x  |  A R x }
)
21eleq2d 2149 . 2  |-  ( A  e.  V  ->  ( B  e.  ( R " { A } )  <-> 
B  e.  { x  |  A R x }
) )
3 brrelex2 4403 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 113 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 3791 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 2745 . . 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 1434   {cab 2068   _Vcvv 2602   {csn 3400   class class class wbr 3787   "cima 4368   Rel wrel 4370
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378
This theorem is referenced by: (None)
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