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Theorem foima 5798
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 5345 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5793 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5733 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 16 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5335 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5796 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2532 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5582   -onto->wfo 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fn 5589  df-f 5590  df-fo 5592
This theorem is referenced by:  foimacnv  5831  domunfican  7789  fiint  7793  fodomfi  7795  cantnflt2  8088  cantnfp1lem3  8095  cantnflt2OLD  8118  cantnfp1lem3OLD  8121  enfin1ai  8760  symgfixelsi  16255  dprdf1o  16869  lmimlbs  18638  cncmp  19658  cmpfi  19674  cnconn  19689  qtopval2  19932  elfm3  20186  rnelfm  20189  fmfnfmlem2  20191  fmfnfm  20194  eupath2  24656  pjordi  26768  qtophaus  27637  ovoliunnfl  29633  voliunnfl  29635  volsupnfl  29636  ismtybndlem  29905  kelac1  30613  gicabl  30651
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