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Theorem imasng 4690
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 2566 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 dfima2 4670 . . 3  |-  ( R
" { A }
)  =  { y  |  E. x  e. 
{ A } x R y }
3 breq1 3767 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
43rexsng 3412 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } x R y  <-> 
A R y ) )
54abbidv 2155 . . 3  |-  ( A  e.  _V  ->  { y  |  E. x  e. 
{ A } x R y }  =  { y  |  A R y } )
62, 5syl5eq 2084 . 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 1243    e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   {csn 3375   class class class wbr 3764   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by:  elreimasng  4691  elimasn  4692  args  4694  fnsnfv  5232  funfvdm2  5237  dfec2  6109  shftfibg  9421  shftfib  9424
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