Theorem foima 6778

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 6055	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6773	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6697	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6045	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6776	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2820	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1559  dom cdm 5643  ran crn 5644   “ cima 5646  –onto→wfo 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-fn 6519  df-f 6520  df-fo 6522

This theorem is referenced by:  foimacnv  6819  fodomfi  9250  domunfican  9260  fiint  9265  cantnflt2  9622  cantnfp1lem3  9629  enfin1ai  10335  symgfixelsi  19466  dprdf1o  20065  lmimlbs  21876  cncmp  23440  cmpfi  23456  cnconn  23470  qtopval2  23744  elfm3  23998  rnelfm  24001  fmfnfmlem2  24003  fmfnfm  24006  eupthvdres  30394  pjordi  32333  qtophaus  34094  poimirlem1  38081  poimirlem2  38082  poimirlem3  38083  poimirlem4  38084  poimirlem5  38085  poimirlem6  38086  poimirlem7  38087  poimirlem9  38089  poimirlem10  38090  poimirlem11  38091  poimirlem12  38092  poimirlem14  38094  poimirlem16  38096  poimirlem17  38097  poimirlem19  38099  poimirlem20  38100  poimirlem22  38102  poimirlem23  38103  poimirlem24  38104  poimirlem25  38105  poimirlem29  38109  poimirlem31  38111  ovoliunnfl  38122  voliunnfl  38124  volsupnfl  38125  ismtybndlem  38266  riccrng1  43100  ricdrng1  43107  kelac1  43601  gicabl  43637  imasubc  49733