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Theorem relimasn 5350
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 4078 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 imaeq2 5323 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( R " { A } )  =  ( R " (/) ) )
31, 2sylbi 195 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  ( R " (/) ) )
4 ima0 5342 . . . . . 6  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2500 . . . . 5  |-  ( -.  A  e.  _V  ->  ( R " { A } )  =  (/) )
65adantl 466 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  (/) )
7 brrelex 5028 . . . . . . 7  |-  ( ( Rel  R  /\  A R y )  ->  A  e.  _V )
87stoic1a 1592 . . . . . 6  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R y )
98nexdv 1870 . . . . 5  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  E. y  A R y )
10 abn0 3790 . . . . . 6  |-  ( { y  |  A R y }  =/=  (/)  <->  E. y  A R y )
1110necon1bbii 2707 . . . . 5  |-  ( -. 
E. y  A R y  <->  { y  |  A R y }  =  (/) )
129, 11sylib 196 . . . 4  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  { y  |  A R y }  =  (/) )
136, 12eqtr4d 2487 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R " { A } )  =  {
y  |  A R y } )
1413ex 434 . 2  |-  ( Rel 
R  ->  ( -.  A  e.  _V  ->  ( R " { A } )  =  {
y  |  A R y } ) )
15 imasng 5349 . 2  |-  ( A  e.  _V  ->  ( R " { A }
)  =  { y  |  A R y } )
1614, 15pm2.61d2 160 1  |-  ( Rel 
R  ->  ( R " { A } )  =  { y  |  A R y } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428   _Vcvv 3095   (/)c0 3770   {csn 4014   class class class wbr 4437   "cima 4992   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002
This theorem is referenced by:  elrelimasn  5351  fnsnfv  5918  funfv2  5926  mapsn  7462  predep  29247  nznngen  31197  nzss  31198  hashnzfz  31201
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