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Theorem imasng 5035
Description: The image of a singleton. (Contributed by NM, 8-May-2005.)
Assertion
Ref Expression
imasng  |-  ( A  e.  B  ->  ( R " { A }
)  =  { y  |  A R y } )
Distinct variable groups:    y, A    y, R
Allowed substitution hint:    B( y)

Proof of Theorem imasng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 dfima2 5014 . . 3  |-  ( R
" { A }
)  =  { y  |  E. x  e. 
{ A } x R y }
3 breq1 4026 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
43rexsng 3673 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } x R y  <-> 
A R y ) )
54abbidv 2397 . . 3  |-  ( A  e.  _V  ->  { y  |  E. x  e. 
{ A } x R y }  =  { y  |  A R y } )
62, 5syl5eq 2327 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
71, 6syl 15 1  |-  ( A  e.  B  ->  ( R " { A }
)  =  { y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   {csn 3640   class class class wbr 4023   "cima 4692
This theorem is referenced by:  relimasn  5036  elimasn  5038  args  5041  dfec2  6663  dfac3  7748  shftfib  11567  areacirclem6  24930
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-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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