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Mirrors > Home > MPE Home > Th. List > foima | Structured version Visualization version GIF version |
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 6090 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 6821 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6747 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
4 | 3 | imaeq2d 6080 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
5 | forn 6824 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | 1, 4, 5 | 3eqtr3a 2799 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 dom cdm 5689 ran crn 5690 “ cima 5692 –onto→wfo 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fn 6566 df-f 6567 df-fo 6569 |
This theorem is referenced by: foimacnv 6866 fodomfi 9348 domunfican 9359 fiint 9364 fiintOLD 9365 fodomfiOLD 9368 cantnflt2 9711 cantnfp1lem3 9718 enfin1ai 10422 symgfixelsi 19468 dprdf1o 20067 lmimlbs 21874 cncmp 23416 cmpfi 23432 cnconn 23446 qtopval2 23720 elfm3 23974 rnelfm 23977 fmfnfmlem2 23979 fmfnfm 23982 eupthvdres 30264 pjordi 32202 qtophaus 33797 poimirlem1 37608 poimirlem2 37609 poimirlem3 37610 poimirlem4 37611 poimirlem5 37612 poimirlem6 37613 poimirlem7 37614 poimirlem9 37616 poimirlem10 37617 poimirlem11 37618 poimirlem12 37619 poimirlem14 37621 poimirlem16 37623 poimirlem17 37624 poimirlem19 37626 poimirlem20 37627 poimirlem22 37629 poimirlem23 37630 poimirlem24 37631 poimirlem25 37632 poimirlem29 37636 poimirlem31 37638 ovoliunnfl 37649 voliunnfl 37651 volsupnfl 37652 ismtybndlem 37793 riccrng1 42508 ricdrng1 42515 kelac1 43052 gicabl 43088 |
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