ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  foima Unicode version

Theorem foima 5251
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 4797 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5246 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5179 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 14 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 4787 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5249 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2145 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   dom cdm 4452   ran crn 4453   "cima 4455   -->wf 5024   -onto->wfo 5026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-fn 5031  df-f 5032  df-fo 5034
This theorem is referenced by:  foimacnv  5284  foima2  5544  fiintim  6693  fidcenumlemr  6718
  Copyright terms: Public domain W3C validator