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| Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) | 
| Ref | Expression | 
|---|---|
| foima | ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | imadmrn 6087 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
| 2 | fof 6819 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
| 3 | 2 | fdmd 6745 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) | 
| 4 | 3 | imaeq2d 6077 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) | 
| 5 | forn 6822 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
| 6 | 1, 4, 5 | 3eqtr3a 2800 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 dom cdm 5684 ran crn 5685 “ cima 5687 –onto→wfo 6558 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fn 6563 df-f 6564 df-fo 6566 | 
| This theorem is referenced by: foimacnv 6864 fodomfi 9351 domunfican 9362 fiint 9367 fiintOLD 9368 fodomfiOLD 9371 cantnflt2 9714 cantnfp1lem3 9721 enfin1ai 10425 symgfixelsi 19454 dprdf1o 20053 lmimlbs 21857 cncmp 23401 cmpfi 23417 cnconn 23431 qtopval2 23705 elfm3 23959 rnelfm 23962 fmfnfmlem2 23964 fmfnfm 23967 eupthvdres 30255 pjordi 32193 qtophaus 33836 poimirlem1 37629 poimirlem2 37630 poimirlem3 37631 poimirlem4 37632 poimirlem5 37633 poimirlem6 37634 poimirlem7 37635 poimirlem9 37637 poimirlem10 37638 poimirlem11 37639 poimirlem12 37640 poimirlem14 37642 poimirlem16 37644 poimirlem17 37645 poimirlem19 37647 poimirlem20 37648 poimirlem22 37650 poimirlem23 37651 poimirlem24 37652 poimirlem25 37653 poimirlem29 37657 poimirlem31 37659 ovoliunnfl 37670 voliunnfl 37672 volsupnfl 37673 ismtybndlem 37814 riccrng1 42536 ricdrng1 42543 kelac1 43080 gicabl 43116 | 
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