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Theorem relimasn 5052
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 3708 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 imaeq2 5024 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( R " { A } )  =  ( R " (/) ) )
31, 2sylbi 187 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  ( R " (/) ) )
4 ima0 5046 . . . . . 6  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2344 . . . . 5  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  (/) )
65adantl 452 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  (/) )
7 brrelex 4743 . . . . . . . 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 1869 . . . . 5  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  E. y  A R y )
11 abn0 3486 . . . . . 6  |-  ( { y  |  A R y }  =/=  (/)  <->  E. y  A R y )
1211necon1bbii 2511 . . . . 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 2331 . . 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 5051 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   (/)c0 3468   {csn 3653   class class class wbr 4039   "cima 4708   Rel wrel 4710
This theorem is referenced by:  elrelimasn  5053  fnsnfv  5598  funfv2  5603  mapsn  6825  predep  24263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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