Theorem foima 6757

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 6035	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6752	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6678	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6025	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6755	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2795	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542  dom cdm 5631  ran crn 5632   “ cima 5634  –onto→wfo 6496

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fn 6501  df-f 6502  df-fo 6504

This theorem is referenced by:  foimacnv  6797  fodomfi  9222  domunfican  9232  fiint  9237  fodomfiOLD  9240  cantnflt2  9594  cantnfp1lem3  9601  enfin1ai  10306  symgfixelsi  19410  dprdf1o  20009  lmimlbs  21816  cncmp  23357  cmpfi  23373  cnconn  23387  qtopval2  23661  elfm3  23915  rnelfm  23918  fmfnfmlem2  23920  fmfnfm  23923  eupthvdres  30305  pjordi  32244  qtophaus  33980  poimirlem1  37942  poimirlem2  37943  poimirlem3  37944  poimirlem4  37945  poimirlem5  37946  poimirlem6  37947  poimirlem7  37948  poimirlem9  37950  poimirlem10  37951  poimirlem11  37952  poimirlem12  37953  poimirlem14  37955  poimirlem16  37957  poimirlem17  37958  poimirlem19  37960  poimirlem20  37961  poimirlem22  37963  poimirlem23  37964  poimirlem24  37965  poimirlem25  37966  poimirlem29  37970  poimirlem31  37972  ovoliunnfl  37983  voliunnfl  37985  volsupnfl  37986  ismtybndlem  38127  riccrng1  42966  ricdrng1  42973  kelac1  43491  gicabl  43527  imasubc  49626