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Theorem imassrn 4983
Description: The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
Assertion
Ref Expression
imassrn  |-  ( A
" B )  C_  ran  A

Proof of Theorem imassrn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exsimpr 1618 . . 3  |-  ( E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
)  ->  E. x <. x ,  y >.  e.  A )
21ss2abi 3229 . 2  |-  { y  |  E. x ( x  e.  B  /\  <.
x ,  y >.  e.  A ) }  C_  { y  |  E. x <. x ,  y >.  e.  A }
3 dfima3 4975 . 2  |-  ( A
" B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
) }
4 dfrn3 4818 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
52, 3, 43sstr4i 3198 1  |-  ( A
" B )  C_  ran  A
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E.wex 1492    e. wcel 2148   {cab 2163    C_ wss 3131   <.cop 3597   ran crn 4629   "cima 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641
This theorem is referenced by:  imaexg  4984  0ima  4990  cnvimass  4993  fimacnv  5647  f1opw2  6079  smores2  6297  ecss  6578  f1imaen2g  6795  fopwdom  6838  ssenen  6853  phplem4dom  6864  isinfinf  6899  fiintim  6930  sbthlem2  6959  sbthlemi3  6960  sbthlemi5  6962  sbthlemi6  6963  ctssdccl  7112  ctinf  12433  ssnnctlemct  12449  mhmima  12880  cnptoprest2  13779  hmeontr  13852  hmeores  13854  tgqioo  14086
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