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Theorem imasng 5357
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 3122 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 dfima2 5337 . . 3  |-  ( R
" { A }
)  =  { y  |  E. x  e. 
{ A } x R y }
3 breq1 4450 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
43rexsng 4063 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } x R y  <-> 
A R y ) )
54abbidv 2603 . . 3  |-  ( A  e.  _V  ->  { y  |  E. x  e. 
{ A } x R y }  =  { y  |  A R y } )
62, 5syl5eq 2520 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
71, 6syl 16 1  |-  ( A  e.  B  ->  ( R " { A }
)  =  { y  |  A R y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   {csn 4027   class class class wbr 4447   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  relimasn  5358  elimasn  5360  args  5363  suppvalbr  6902  dfec2  7311  dfac3  8498  shftfib  12864  areacirclem5  29688
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