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Theorem elrelimasn 5181
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 5180 . . 3  |-  ( Rel 
R  ->  ( R " { A } )  =  { x  |  A R x }
)
21eleq2d 2472 . 2  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  B  e.  { x  |  A R x } ) )
3 brrelex2 4863 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 432 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 4399 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 3202 . . 3  |-  ( ( A R B  ->  B  e.  _V )  ->  ( B  e.  {
x  |  A R x }  <->  A R B ) )
74, 6syl 17 . 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 1842   {cab 2387   _Vcvv 3059   {csn 3972   class class class wbr 4395   "cima 4826   Rel wrel 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836
This theorem is referenced by:  eliniseg2  5197  dprd2dlem2  17409  dprd2dlem1  17410  dprd2da  17411  dprd2d2  17413  dpjfval  17424  ustuqtop4  21039  utop2nei  21045  utop3cls  21046  ucncn  21080  cnambfre  31435  frege133d  35744  nzin  36071
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