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Theorem imasng 5138
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 2881 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 dfima2 5117 . . 3  |-  ( R
" { A }
)  =  { y  |  E. x  e. 
{ A } x R y }
3 breq1 4128 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
43rexsng 3762 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } x R y  <-> 
A R y ) )
54abbidv 2480 . . 3  |-  ( A  e.  _V  ->  { y  |  E. x  e. 
{ A } x R y }  =  { y  |  A R y } )
62, 5syl5eq 2410 . 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 1647    e. wcel 1715   {cab 2352   E.wrex 2629   _Vcvv 2873   {csn 3729   class class class wbr 4125   "cima 4795
This theorem is referenced by:  relimasn  5139  elimasn  5141  args  5144  dfec2  6805  dfac3  7895  shftfib  11774  areacirclem6  25705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805
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