Theorem foima 6693

Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Assertion

Ref	Expression
foima	⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Proof of Theorem foima

Step	Hyp	Ref	Expression
1		imadmrn 5979	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6688	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6611	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 5969	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6691	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2802	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1539  dom cdm 5589  ran crn 5590   “ cima 5592  –onto→wfo 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fn 6436  df-f 6437  df-fo 6439

This theorem is referenced by:  foimacnv  6733  domunfican  9087  fiint  9091  fodomfi  9092  cantnflt2  9431  cantnfp1lem3  9438  enfin1ai  10140  symgfixelsi  19043  dprdf1o  19635  lmimlbs  21043  cncmp  22543  cmpfi  22559  cnconn  22573  qtopval2  22847  elfm3  23101  rnelfm  23104  fmfnfmlem2  23106  fmfnfm  23109  eupthvdres  28599  pjordi  30535  qtophaus  31786  poimirlem1  35778  poimirlem2  35779  poimirlem3  35780  poimirlem4  35781  poimirlem5  35782  poimirlem6  35783  poimirlem7  35784  poimirlem9  35786  poimirlem10  35787  poimirlem11  35788  poimirlem12  35789  poimirlem14  35791  poimirlem16  35793  poimirlem17  35794  poimirlem19  35796  poimirlem20  35797  poimirlem22  35799  poimirlem23  35800  poimirlem24  35801  poimirlem25  35802  poimirlem29  35806  poimirlem31  35808  ovoliunnfl  35819  voliunnfl  35821  volsupnfl  35822  ismtybndlem  35964  kelac1  40888  gicabl  40924