Theorem foima 6595

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 5939	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6590	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6523	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 5929	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6593	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2880	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1537  dom cdm 5555  ran crn 5556   “ cima 5558  –onto→wfo 6353

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 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-fn 6358  df-f 6359  df-fo 6361

This theorem is referenced by:  foimacnv  6632  domunfican  8791  fiint  8795  fodomfi  8797  cantnflt2  9136  cantnfp1lem3  9143  enfin1ai  9806  symgfixelsi  18563  dprdf1o  19154  lmimlbs  20980  cncmp  22000  cmpfi  22016  cnconn  22030  qtopval2  22304  elfm3  22558  rnelfm  22561  fmfnfmlem2  22563  fmfnfm  22566  eupthvdres  28014  pjordi  29950  qtophaus  31100  poimirlem1  34908  poimirlem2  34909  poimirlem3  34910  poimirlem4  34911  poimirlem5  34912  poimirlem6  34913  poimirlem7  34914  poimirlem9  34916  poimirlem10  34917  poimirlem11  34918  poimirlem12  34919  poimirlem14  34921  poimirlem16  34923  poimirlem17  34924  poimirlem19  34926  poimirlem20  34927  poimirlem22  34929  poimirlem23  34930  poimirlem24  34931  poimirlem25  34932  poimirlem29  34936  poimirlem31  34938  ovoliunnfl  34949  voliunnfl  34951  volsupnfl  34952  ismtybndlem  35099  kelac1  39683  gicabl  39719