Theorem foima 6751

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 6029	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6746	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6672	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6019	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6749	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2796	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542  dom cdm 5624  ran crn 5625   “ cima 5627  –onto→wfo 6490

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fn 6495  df-f 6496  df-fo 6498

This theorem is referenced by:  foimacnv  6791  fodomfi  9215  domunfican  9225  fiint  9230  fodomfiOLD  9233  cantnflt2  9585  cantnfp1lem3  9592  enfin1ai  10297  symgfixelsi  19401  dprdf1o  20000  lmimlbs  21826  cncmp  23367  cmpfi  23383  cnconn  23397  qtopval2  23671  elfm3  23925  rnelfm  23928  fmfnfmlem2  23930  fmfnfm  23933  eupthvdres  30320  pjordi  32259  qtophaus  33996  poimirlem1  37956  poimirlem2  37957  poimirlem3  37958  poimirlem4  37959  poimirlem5  37960  poimirlem6  37961  poimirlem7  37962  poimirlem9  37964  poimirlem10  37965  poimirlem11  37966  poimirlem12  37967  poimirlem14  37969  poimirlem16  37971  poimirlem17  37972  poimirlem19  37974  poimirlem20  37975  poimirlem22  37977  poimirlem23  37978  poimirlem24  37979  poimirlem25  37980  poimirlem29  37984  poimirlem31  37986  ovoliunnfl  37997  voliunnfl  37999  volsupnfl  38000  ismtybndlem  38141  riccrng1  42980  ricdrng1  42987  kelac1  43509  gicabl  43545  imasubc  49638