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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 4162 . 2  |-  ( ( ( A " B
)  C_  ran  A  /\  ran  A  e.  _V )  ->  ( A " B
)  e.  _V )
41, 2, 3sylancr 646 1  |-  ( A  e.  V  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1685   _Vcvv 2790    C_ wss 3154   ran crn 4690   "cima 4692
This theorem is referenced by:  frxp  6187  ecexg  6660  pw2f1o  6963  fopwdom  6966  ssenen  7031  fiint  7129  fissuni  7156  fipreima  7157  marypha1lem  7182  cantnfdm  7361  cantnfcl  7364  cantnfval  7365  cantnflt2  7370  cantnff  7371  cantnflem1  7387  cnfcom2  7401  cnfcom3lem  7402  cnfcom3  7403  infxpenlem  7637  ackbij2lem2  7862  enfin2i  7943  fin1a2lem7  8028  fpwwe  8264  canthwelem  8268  tskuni  8401  isacs4lem  14266  gsumvalx  14446  gicsubgen  14737  gsumzaddlem  15198  gsum2d  15218  isunit  15434  ptbasfi  17271  xkococnlem  17348  qtopval  17381  hmphdis  17482  nghmfval  18226  cnheiborlem  18447  itg2gt0  19110  fta1glem2  19547  fta1blem  19549  lgsqrlem4  20578  shsval  21884  nlfnval  22454  ballotlemrval  23070  ballotlem7  23088  dfrdg2  23554  brapply  23885  dfrdg4  23896  intopcoaconb  24940  intopcoaconc  24941  prcnt  24951  nds  25550  tailval  25722  isnacs3  26185  pw2f1ocnv  26530  pw2f1o2val  26532  lmhmlnmsplit  26585  lkrval  28546
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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