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Theorem relimasn 5036
Description: The image of a singleton. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
relimasn  |-  ( Rel 
R  ->  ( R " { A } )  =  { y  |  A R y } )
Distinct variable groups:    y, A    y, R

Proof of Theorem relimasn
StepHypRef Expression
1 snprc 3695 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 imaeq2 5008 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( R " { A } )  =  ( R " (/) ) )
31, 2sylbi 187 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  ( R " (/) ) )
4 ima0 5030 . . . . . 6  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2331 . . . . 5  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  (/) )
65adantl 452 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  (/) )
7 brrelex 4727 . . . . . . . 8  |-  ( ( Rel  R  /\  A R y )  ->  A  e.  _V )
87ex 423 . . . . . . 7  |-  ( Rel 
R  ->  ( A R y  ->  A  e.  _V ) )
98con3and 428 . . . . . 6  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R y )
109nexdv 1857 . . . . 5  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  E. y  A R y )
11 abn0 3473 . . . . . 6  |-  ( { y  |  A R y }  =/=  (/)  <->  E. y  A R y )
1211necon1bbii 2498 . . . . 5  |-  ( -. 
E. y  A R y  <->  { y  |  A R y }  =  (/) )
1310, 12sylib 188 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  { y  |  A R y }  =  (/) )
146, 13eqtr4d 2318 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  {
y  |  A R y } )
1514ex 423 . 2  |-  ( Rel 
R  ->  ( -.  A  e.  _V  ->  ( R " { A } )  =  {
y  |  A R y } ) )
16 imasng 5035 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
1715, 16pm2.61d2 152 1  |-  ( Rel 
R  ->  ( R " { A } )  =  { y  |  A R y } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   (/)c0 3455   {csn 3640   class class class wbr 4023   "cima 4692   Rel wrel 4694
This theorem is referenced by:  elrelimasn  5037  fnsnfv  5582  funfv2  5587  mapsn  6809  predep  24192
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-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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