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Theorem elrelimasn 5214
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 5213 . . 3  |-  ( Rel 
R  ->  ( R " { A } )  =  { x  |  A R x }
)
21eleq2d 2510 . 2  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  B  e.  { x  |  A R x } ) )
3 brrelex2 4899 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 434 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 4317 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 3133 . . 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 253 1  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  A R B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756   {cab 2429   _Vcvv 2993   {csn 3898   class class class wbr 4313   "cima 4864   Rel wrel 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-xp 4867  df-rel 4868  df-cnv 4869  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874
This theorem is referenced by:  eliniseg2  5229  dprd2dlem2  16561  dprd2dlem1  16562  dprd2da  16563  dprd2d2  16565  dpjfval  16576  ustuqtop4  19841  utop2nei  19847  utop3cls  19848  ucncn  19882  cnambfre  28466
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