Theorem foima 6597

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 5941	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6592	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6525	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 5931	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6595	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2882	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1537  dom cdm 5557  ran crn 5558   “ cima 5560  –onto→wfo 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-fn 6360  df-f 6361  df-fo 6363

This theorem is referenced by:  foimacnv  6634  domunfican  8793  fiint  8797  fodomfi  8799  cantnflt2  9138  cantnfp1lem3  9145  enfin1ai  9808  symgfixelsi  18565  dprdf1o  19156  lmimlbs  20982  cncmp  22002  cmpfi  22018  cnconn  22032  qtopval2  22306  elfm3  22560  rnelfm  22563  fmfnfmlem2  22565  fmfnfm  22568  eupthvdres  28016  pjordi  29952  qtophaus  31102  poimirlem1  34895  poimirlem2  34896  poimirlem3  34897  poimirlem4  34898  poimirlem5  34899  poimirlem6  34900  poimirlem7  34901  poimirlem9  34903  poimirlem10  34904  poimirlem11  34905  poimirlem12  34906  poimirlem14  34908  poimirlem16  34910  poimirlem17  34911  poimirlem19  34913  poimirlem20  34914  poimirlem22  34916  poimirlem23  34917  poimirlem24  34918  poimirlem25  34919  poimirlem29  34923  poimirlem31  34925  ovoliunnfl  34936  voliunnfl  34938  volsupnfl  34939  ismtybndlem  35086  kelac1  39670  gicabl  39706