MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrelimasn Unicode version

Theorem elrelimasn 5037
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 5036 . . 3  |-  ( Rel 
R  ->  ( R " { A } )  =  { x  |  A R x }
)
21eleq2d 2350 . 2  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  B  e.  { x  |  A R x } ) )
3 brrelex2 4728 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 423 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 4027 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 2920 . . 3  |-  ( ( A R B  ->  B  e.  _V )  ->  ( B  e.  {
x  |  A R x }  <->  A R B ) )
74, 6syl 15 . 2  |-  ( Rel 
R  ->  ( B  e.  { x  |  A R x }  <->  A R B ) )
82, 7bitrd 244 1  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  A R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   {cab 2269   _Vcvv 2788   {csn 3640   class class class wbr 4023   "cima 4692   Rel wrel 4694
This theorem is referenced by:  eliniseg2  5053  dprd2dlem2  15275  dprd2dlem1  15276  dprd2da  15277  dprd2d2  15279  dpjfval  15290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
  Copyright terms: Public domain W3C validator