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Theorem foima 6591
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 5936 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6586 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
32fdmd 6519 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
43imaeq2d 5926 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
5 forn 6589 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
61, 4, 53eqtr3a 2884 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  dom cdm 5553  ran crn 5554  cima 5556  ontowfo 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-fn 6354  df-f 6355  df-fo 6357
This theorem is referenced by:  foimacnv  6628  domunfican  8783  fiint  8787  fodomfi  8789  cantnflt2  9128  cantnfp1lem3  9135  enfin1ai  9798  symgfixelsi  18485  dprdf1o  19076  lmimlbs  20896  cncmp  21916  cmpfi  21932  cnconn  21946  qtopval2  22220  elfm3  22474  rnelfm  22477  fmfnfmlem2  22479  fmfnfm  22482  eupthvdres  27928  pjordi  29864  qtophaus  30986  poimirlem1  34760  poimirlem2  34761  poimirlem3  34762  poimirlem4  34763  poimirlem5  34764  poimirlem6  34765  poimirlem7  34766  poimirlem9  34768  poimirlem10  34769  poimirlem11  34770  poimirlem12  34771  poimirlem14  34773  poimirlem16  34775  poimirlem17  34776  poimirlem19  34778  poimirlem20  34779  poimirlem22  34781  poimirlem23  34782  poimirlem24  34783  poimirlem25  34784  poimirlem29  34788  poimirlem31  34790  ovoliunnfl  34801  voliunnfl  34803  volsupnfl  34804  ismtybndlem  34952  kelac1  39524  gicabl  39560
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