Theorem foima 6570

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 5906	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6565	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6497	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 5896	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6568	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2857	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1538  dom cdm 5519  ran crn 5520   “ cima 5522  –onto→wfo 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fn 6327  df-f 6328  df-fo 6330

This theorem is referenced by:  foimacnv  6607  domunfican  8775  fiint  8779  fodomfi  8781  cantnflt2  9120  cantnfp1lem3  9127  enfin1ai  9795  symgfixelsi  18555  dprdf1o  19147  lmimlbs  20525  cncmp  21997  cmpfi  22013  cnconn  22027  qtopval2  22301  elfm3  22555  rnelfm  22558  fmfnfmlem2  22560  fmfnfm  22563  eupthvdres  28020  pjordi  29956  qtophaus  31189  poimirlem1  35058  poimirlem2  35059  poimirlem3  35060  poimirlem4  35061  poimirlem5  35062  poimirlem6  35063  poimirlem7  35064  poimirlem9  35066  poimirlem10  35067  poimirlem11  35068  poimirlem12  35069  poimirlem14  35071  poimirlem16  35073  poimirlem17  35074  poimirlem19  35076  poimirlem20  35077  poimirlem22  35079  poimirlem23  35080  poimirlem24  35081  poimirlem25  35082  poimirlem29  35086  poimirlem31  35088  ovoliunnfl  35099  voliunnfl  35101  volsupnfl  35102  ismtybndlem  35244  kelac1  40007  gicabl  40043