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Theorem foima 5622
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 5176 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5617 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5560 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 16 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5166 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5620 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2497 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   dom cdm 4836   ran crn 4837   "cima 4839   -->wf 5411   -onto->wfo 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fn 5418  df-f 5419  df-fo 5421
This theorem is referenced by:  foimacnv  5655  domunfican  7580  fiint  7584  fodomfi  7586  cantnflt2  7877  cantnfp1lem3  7884  cantnflt2OLD  7907  cantnfp1lem3OLD  7910  enfin1ai  8549  symgfixelsi  15933  dprdf1o  16519  lmimlbs  18224  cncmp  18954  cmpfi  18970  cnconn  18985  qtopval2  19228  elfm3  19482  rnelfm  19485  fmfnfmlem2  19487  fmfnfm  19490  eupath2  23536  pjordi  25512  ovoliunnfl  28358  voliunnfl  28360  volsupnfl  28361  ismtybndlem  28630  kelac1  29341  gicabl  29379
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