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Theorem imasng 4766
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 2624 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 dfima2 4745 . . 3  |-  ( R
" { A }
)  =  { y  |  E. x  e. 
{ A } x R y }
3 breq1 3825 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
43rexsng 3469 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } x R y  <-> 
A R y ) )
54abbidv 2202 . . 3  |-  ( A  e.  _V  ->  { y  |  E. x  e. 
{ A } x R y }  =  { y  |  A R y } )
62, 5syl5eq 2129 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
71, 6syl 14 1  |-  ( A  e.  B  ->  ( R " { A }
)  =  { y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   {cab 2071   E.wrex 2356   _Vcvv 2615   {csn 3431   class class class wbr 3822   "cima 4416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by:  elreimasng  4767  elimasn  4768  args  4770  fnsnfv  5328  funfvdm2  5333  dfec2  6249  mapsn  6401  shftfibg  10154  shftfib  10157
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