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Theorem imassrn 4981
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 3227 . 2  |-  { y  |  E. x ( x  e.  B  /\  <.
x ,  y >.  e.  A ) }  C_  { y  |  E. x <. x ,  y >.  e.  A }
3 dfima3 4973 . 2  |-  ( A
" B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
) }
4 dfrn3 4816 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
52, 3, 43sstr4i 3196 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 3129   <.cop 3595   ran crn 4627   "cima 4629
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  imaexg  4982  0ima  4988  cnvimass  4991  fimacnv  5645  f1opw2  6076  smores2  6294  ecss  6575  f1imaen2g  6792  fopwdom  6835  ssenen  6850  phplem4dom  6861  isinfinf  6896  fiintim  6927  sbthlem2  6956  sbthlemi3  6957  sbthlemi5  6959  sbthlemi6  6960  ctssdccl  7109  ctinf  12430  ssnnctlemct  12446  mhmima  12874  cnptoprest2  13710  hmeontr  13783  hmeores  13785  tgqioo  14017
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