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Theorem foima 6826
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 6090 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6821 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
32fdmd 6747 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
43imaeq2d 6080 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
5 forn 6824 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
61, 4, 53eqtr3a 2799 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  dom cdm 5689  ran crn 5690  cima 5692  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fn 6566  df-f 6567  df-fo 6569
This theorem is referenced by:  foimacnv  6866  fodomfi  9348  domunfican  9359  fiint  9364  fiintOLD  9365  fodomfiOLD  9368  cantnflt2  9711  cantnfp1lem3  9718  enfin1ai  10422  symgfixelsi  19468  dprdf1o  20067  lmimlbs  21874  cncmp  23416  cmpfi  23432  cnconn  23446  qtopval2  23720  elfm3  23974  rnelfm  23977  fmfnfmlem2  23979  fmfnfm  23982  eupthvdres  30264  pjordi  32202  qtophaus  33797  poimirlem1  37608  poimirlem2  37609  poimirlem3  37610  poimirlem4  37611  poimirlem5  37612  poimirlem6  37613  poimirlem7  37614  poimirlem9  37616  poimirlem10  37617  poimirlem11  37618  poimirlem12  37619  poimirlem14  37621  poimirlem16  37623  poimirlem17  37624  poimirlem19  37626  poimirlem20  37627  poimirlem22  37629  poimirlem23  37630  poimirlem24  37631  poimirlem25  37632  poimirlem29  37636  poimirlem31  37638  ovoliunnfl  37649  voliunnfl  37651  volsupnfl  37652  ismtybndlem  37793  riccrng1  42508  ricdrng1  42515  kelac1  43052  gicabl  43088
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