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Theorem elrelimasn 5352
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 5351 . . 3  |-  ( Rel 
R  ->  ( R " { A } )  =  { x  |  A R x }
)
21eleq2d 2530 . 2  |-  ( Rel 
R  ->  ( B  e.  ( R " { A } )  <->  B  e.  { x  |  A R x } ) )
3 brrelex2 5031 . . . 4  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
43ex 434 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  B  e. 
_V ) )
5 breq2 4444 . . . 4  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
65elab3g 3249 . . 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 1762   {cab 2445   _Vcvv 3106   {csn 4020   class class class wbr 4440   "cima 4995   Rel wrel 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  eliniseg2  5367  dprd2dlem2  16872  dprd2dlem1  16873  dprd2da  16874  dprd2d2  16876  dpjfval  16887  ustuqtop4  20475  utop2nei  20481  utop3cls  20482  ucncn  20516  cnambfre  29627
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