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Theorem relimasn 5229
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 3873 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 imaeq2 5201 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( R " { A } )  =  ( R " (/) ) )
31, 2sylbi 189 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  ( R " (/) ) )
4 ima0 5223 . . . . . 6  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2486 . . . . 5  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  (/) )
65adantl 454 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  (/) )
7 brrelex 4918 . . . . . . . 8  |-  ( ( Rel  R  /\  A R y )  ->  A  e.  _V )
87ex 425 . . . . . . 7  |-  ( Rel 
R  ->  ( A R y  ->  A  e.  _V ) )
98con3and 430 . . . . . 6  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R y )
109nexdv 1942 . . . . 5  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  E. y  A R y )
11 abn0 3648 . . . . . 6  |-  ( { y  |  A R y }  =/=  (/)  <->  E. y  A R y )
1211necon1bbii 2658 . . . . 5  |-  ( -. 
E. y  A R y  <->  { y  |  A R y }  =  (/) )
1310, 12sylib 190 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  { y  |  A R y }  =  (/) )
146, 13eqtr4d 2473 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  {
y  |  A R y } )
1514ex 425 . 2  |-  ( Rel 
R  ->  ( -.  A  e.  _V  ->  ( R " { A } )  =  {
y  |  A R y } ) )
16 imasng 5228 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
1715, 16pm2.61d2 155 1  |-  ( Rel 
R  ->  ( R " { A } )  =  { y  |  A R y } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958   (/)c0 3630   {csn 3816   class class class wbr 4214   "cima 4883   Rel wrel 4885
This theorem is referenced by:  elrelimasn  5230  fnsnfv  5788  funfv2  5793  mapsn  7057  predep  25469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893
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