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Theorem foima 6824
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 6087 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6819 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
32fdmd 6745 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
43imaeq2d 6077 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
5 forn 6822 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
61, 4, 53eqtr3a 2800 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  dom cdm 5684  ran crn 5685  cima 5687  ontowfo 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fn 6563  df-f 6564  df-fo 6566
This theorem is referenced by:  foimacnv  6864  fodomfi  9351  domunfican  9362  fiint  9367  fiintOLD  9368  fodomfiOLD  9371  cantnflt2  9714  cantnfp1lem3  9721  enfin1ai  10425  symgfixelsi  19454  dprdf1o  20053  lmimlbs  21857  cncmp  23401  cmpfi  23417  cnconn  23431  qtopval2  23705  elfm3  23959  rnelfm  23962  fmfnfmlem2  23964  fmfnfm  23967  eupthvdres  30255  pjordi  32193  qtophaus  33836  poimirlem1  37629  poimirlem2  37630  poimirlem3  37631  poimirlem4  37632  poimirlem5  37633  poimirlem6  37634  poimirlem7  37635  poimirlem9  37637  poimirlem10  37638  poimirlem11  37639  poimirlem12  37640  poimirlem14  37642  poimirlem16  37644  poimirlem17  37645  poimirlem19  37647  poimirlem20  37648  poimirlem22  37650  poimirlem23  37651  poimirlem24  37652  poimirlem25  37653  poimirlem29  37657  poimirlem31  37659  ovoliunnfl  37670  voliunnfl  37672  volsupnfl  37673  ismtybndlem  37814  riccrng1  42536  ricdrng1  42543  kelac1  43080  gicabl  43116
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