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Theorem foima 6261
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 5617 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6256 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
3 fdm 6191 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
42, 3syl 17 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
54imaeq2d 5607 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
6 forn 6259 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
71, 5, 63eqtr3a 2828 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  dom cdm 5249  ran crn 5250  cima 5252  wf 6027  ontowfo 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fn 6034  df-f 6035  df-fo 6037
This theorem is referenced by:  foimacnv  6295  domunfican  8388  fiint  8392  fodomfi  8394  cantnflt2  8733  cantnfp1lem3  8740  enfin1ai  9407  symgfixelsi  18061  dprdf1o  18638  lmimlbs  20391  cncmp  21415  cmpfi  21431  cnconn  21445  qtopval2  21719  elfm3  21973  rnelfm  21976  fmfnfmlem2  21978  fmfnfm  21981  eupthvdres  27412  pjordi  29366  qtophaus  30237  poimirlem1  33736  poimirlem2  33737  poimirlem3  33738  poimirlem4  33739  poimirlem5  33740  poimirlem6  33741  poimirlem7  33742  poimirlem9  33744  poimirlem10  33745  poimirlem11  33746  poimirlem12  33747  poimirlem14  33749  poimirlem16  33751  poimirlem17  33752  poimirlem19  33754  poimirlem20  33755  poimirlem22  33757  poimirlem23  33758  poimirlem24  33759  poimirlem25  33760  poimirlem29  33764  poimirlem31  33766  ovoliunnfl  33777  voliunnfl  33779  volsupnfl  33780  ismtybndlem  33930  kelac1  38152  gicabl  38188
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