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Theorem elrelimasn 5229
Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
elrelimasn  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  A R B ) )

Proof of Theorem elrelimasn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 relimasn 5228 . . 3  |-  ( Rel 
R  ->  ( R " { A } )  =  { x  |  A R x }
)
21eleq2d 2504 . 2  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  B  e.  { x  |  A R x } ) )
3 brrelex2 4918 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 425 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 4217 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 3089 . . 3  |-  ( ( A R B  ->  B  e.  _V )  ->  ( B  e.  {
x  |  A R x }  <->  A R B ) )
74, 6syl 16 . 2  |-  ( Rel 
R  ->  ( B  e.  { x  |  A R x }  <->  A R B ) )
82, 7bitrd 246 1  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  A R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   {cab 2423   _Vcvv 2957   {csn 3815   class class class wbr 4213   "cima 4882   Rel wrel 4884
This theorem is referenced by:  eliniseg2  5245  dprd2dlem2  15599  dprd2dlem1  15600  dprd2da  15601  dprd2d2  15603  dpjfval  15614  ustuqtop4  18275  utop2nei  18281  utop3cls  18282  ucncn  18316  cnambfre  26256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892
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