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Theorem relimasn 5210
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 4063 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 imaeq2 5183 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( R " { A } )  =  ( R " (/) ) )
31, 2sylbi 198 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  ( R " (/) ) )
4 ima0 5202 . . . . . 6  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2479 . . . . 5  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  (/) )
65adantl 467 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  (/) )
7 brrelex 4892 . . . . . . 7  |-  ( ( Rel  R  /\  A R y )  ->  A  e.  _V )
87stoic1a 1649 . . . . . 6  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R y )
98nexdv 1775 . . . . 5  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  E. y  A R y )
10 abn0 3781 . . . . . 6  |-  ( { y  |  A R y }  =/=  (/)  <->  E. y  A R y )
1110necon1bbii 2684 . . . . 5  |-  ( -. 
E. y  A R y  <->  { y  |  A R y }  =  (/) )
129, 11sylib 199 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  { y  |  A R y }  =  (/) )
136, 12eqtr4d 2466 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  {
y  |  A R y } )
1413ex 435 . 2  |-  ( Rel 
R  ->  ( -.  A  e.  _V  ->  ( R " { A } )  =  {
y  |  A R y } ) )
15 imasng 5209 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
1614, 15pm2.61d2 163 1  |-  ( Rel 
R  ->  ( R " { A } )  =  { y  |  A R y } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   _Vcvv 3080   (/)c0 3761   {csn 3998   class class class wbr 4423   "cima 4856   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by:  elrelimasn  5211  predep  5425  fnsnfv  5942  funfv2  5950  mapsn  7525  nznngen  36636  nzss  36637  hashnzfz  36640
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