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Theorem foima 6033
 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 5395 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 6028 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
3 fdm 5964 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
42, 3syl 17 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
54imaeq2d 5385 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
6 forn 6031 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
71, 5, 63eqtr3a 2668 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  dom cdm 5038  ran crn 5039   “ cima 5041  ⟶wf 5800  –onto→wfo 5802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fn 5807  df-f 5808  df-fo 5810 This theorem is referenced by:  foimacnv  6067  domunfican  8118  fiint  8122  fodomfi  8124  cantnflt2  8453  cantnfp1lem3  8460  enfin1ai  9089  symgfixelsi  17678  dprdf1o  18254  lmimlbs  19994  cncmp  21005  cmpfi  21021  cnconn  21035  qtopval2  21309  elfm3  21564  rnelfm  21567  fmfnfmlem2  21569  fmfnfm  21572  eupath2  26507  pjordi  28416  qtophaus  29231  poimirlem1  32580  poimirlem2  32581  poimirlem3  32582  poimirlem4  32583  poimirlem5  32584  poimirlem6  32585  poimirlem7  32586  poimirlem9  32588  poimirlem10  32589  poimirlem11  32590  poimirlem12  32591  poimirlem14  32593  poimirlem16  32595  poimirlem17  32596  poimirlem19  32598  poimirlem20  32599  poimirlem22  32601  poimirlem23  32602  poimirlem24  32603  poimirlem25  32604  poimirlem29  32608  poimirlem31  32610  ovoliunnfl  32621  voliunnfl  32623  volsupnfl  32624  ismtybndlem  32775  kelac1  36651  gicabl  36687  eupthvdres  41403
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