Theorem foima 6799

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 6074	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6794	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6718	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6064	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6797	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2822	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1570  dom cdm 5663  ran crn 5664   “ cima 5666  –onto→wfo 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fn 6541  df-f 6542  df-fo 6544

This theorem is referenced by:  foimacnv  6840  fodomfi  9273  domunfican  9282  fiint  9287  cantnflt2  9643  cantnfp1lem3  9650  enfin1ai  10369  symgfixelsi  19506  dprdf1o  20105  lmimlbs  21967  cncmp  23530  cmpfi  23546  cnconn  23560  qtopval2  23834  elfm3  24088  rnelfm  24091  fmfnfmlem2  24093  fmfnfm  24096  eupthvdres  30567  pjordi  32506  qtophaus  34207  poimirlem1  38253  poimirlem2  38254  poimirlem3  38255  poimirlem4  38256  poimirlem5  38257  poimirlem6  38258  poimirlem7  38259  poimirlem9  38261  poimirlem10  38262  poimirlem11  38263  poimirlem12  38264  poimirlem14  38266  poimirlem16  38268  poimirlem17  38269  poimirlem19  38271  poimirlem20  38272  poimirlem22  38274  poimirlem23  38275  poimirlem24  38276  poimirlem25  38277  poimirlem29  38281  poimirlem31  38283  ovoliunnfl  38294  voliunnfl  38296  volsupnfl  38297  ismtybndlem  38438  riccrng1  43272  ricdrng1  43279  kelac1  43773  gicabl  43809  imasubc  49912