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Theorem elreimasng 4691
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 4690 . . 3  |-  ( A  e.  V  ->  ( R " { A }
)  =  { x  |  A R x }
)
21eleq2d 2107 . 2  |-  ( A  e.  V  ->  ( B  e.  ( R " { A } )  <-> 
B  e.  { x  |  A R x }
) )
3 brrelex2 4383 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 108 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 3768 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 2693 . . 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 436 1  |-  ( ( Rel  R  /\  A  e.  V )  ->  ( B  e.  ( R " { A } )  <-> 
A R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    e. wcel 1393   {cab 2026   _Vcvv 2557   {csn 3375   class class class wbr 3764   "cima 4348   Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by: (None)
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