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Theorem imaexg 5026
Description: The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
Assertion
Ref Expression
imaexg  |-  ( A  e.  V  ->  ( A " B )  e. 
_V )

Proof of Theorem imaexg
StepHypRef Expression
1 imassrn 5025 . 2  |-  ( A
" B )  C_  ran  A
2 rnexg 4940 . 2  |-  ( A  e.  V  ->  ran  A  e.  _V )
3 ssexg 4160 . 2  |-  ( ( ( A " B
)  C_  ran  A  /\  ran  A  e.  _V )  ->  ( A " B
)  e.  _V )
41, 2, 3sylancr 644 1  |-  ( A  e.  V  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ran crn 4690   "cima 4692
This theorem is referenced by:  frxp  6225  ecexg  6664  pw2f1o  6967  fopwdom  6970  ssenen  7035  fiint  7133  fissuni  7160  fipreima  7161  marypha1lem  7186  cantnfdm  7365  cantnfcl  7368  cantnfval  7369  cantnflt2  7374  cantnff  7375  cantnflem1  7391  cnfcom2  7405  cnfcom3lem  7406  cnfcom3  7407  infxpenlem  7641  ackbij2lem2  7866  enfin2i  7947  fin1a2lem7  8032  fpwwe  8268  canthwelem  8272  tskuni  8405  isacs4lem  14271  gsumvalx  14451  gicsubgen  14742  gsumzaddlem  15203  gsum2d  15223  isunit  15439  ptbasfi  17276  xkococnlem  17353  qtopval  17386  hmphdis  17487  nghmfval  18231  cnheiborlem  18452  itg2gt0  19115  fta1glem2  19552  fta1blem  19554  lgsqrlem4  20583  shsval  21891  nlfnval  22461  ballotlemrval  23076  ballotlem7  23094  mbfmcnt  23573  orvcval  23658  coinfliprv  23683  dfrdg2  24152  brapply  24477  dfrdg4  24488  intopcoaconb  25540  intopcoaconc  25541  prcnt  25551  nds  26150  tailval  26322  isnacs3  26785  pw2f1ocnv  27130  pw2f1o2val  27132  lmhmlnmsplit  27185  lkrval  29278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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