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Theorem foima 6077
 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 5435 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6072 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
3 fdm 6008 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
42, 3syl 17 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
54imaeq2d 5425 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
6 forn 6075 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
71, 5, 63eqtr3a 2679 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480  dom cdm 5074  ran crn 5075   “ cima 5077  ⟶wf 5843  –onto→wfo 5845 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fn 5850  df-f 5851  df-fo 5853 This theorem is referenced by:  foimacnv  6111  domunfican  8177  fiint  8181  fodomfi  8183  cantnflt2  8514  cantnfp1lem3  8521  enfin1ai  9150  symgfixelsi  17776  dprdf1o  18352  lmimlbs  20094  cncmp  21105  cmpfi  21121  cnconn  21135  qtopval2  21409  elfm3  21664  rnelfm  21667  fmfnfmlem2  21669  fmfnfm  21672  eupthvdres  26961  pjordi  28881  qtophaus  29685  poimirlem1  33042  poimirlem2  33043  poimirlem3  33044  poimirlem4  33045  poimirlem5  33046  poimirlem6  33047  poimirlem7  33048  poimirlem9  33050  poimirlem10  33051  poimirlem11  33052  poimirlem12  33053  poimirlem14  33055  poimirlem16  33057  poimirlem17  33058  poimirlem19  33060  poimirlem20  33061  poimirlem22  33063  poimirlem23  33064  poimirlem24  33065  poimirlem25  33066  poimirlem29  33070  poimirlem31  33072  ovoliunnfl  33083  voliunnfl  33085  volsupnfl  33086  ismtybndlem  33237  kelac1  37113  gicabl  37149
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