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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 5395 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 6028 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | fdm 5964 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
5 | 4 | imaeq2d 5385 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
6 | forn 6031 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
7 | 1, 5, 6 | 3eqtr3a 2668 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 –onto→wfo 5802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fn 5807 df-f 5808 df-fo 5810 |
This theorem is referenced by: foimacnv 6067 domunfican 8118 fiint 8122 fodomfi 8124 cantnflt2 8453 cantnfp1lem3 8460 enfin1ai 9089 symgfixelsi 17678 dprdf1o 18254 lmimlbs 19994 cncmp 21005 cmpfi 21021 cnconn 21035 qtopval2 21309 elfm3 21564 rnelfm 21567 fmfnfmlem2 21569 fmfnfm 21572 eupath2 26507 pjordi 28416 qtophaus 29231 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem5 32584 poimirlem6 32585 poimirlem7 32586 poimirlem9 32588 poimirlem10 32589 poimirlem11 32590 poimirlem12 32591 poimirlem14 32593 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem22 32601 poimirlem23 32602 poimirlem24 32603 poimirlem25 32604 poimirlem29 32608 poimirlem31 32610 ovoliunnfl 32621 voliunnfl 32623 volsupnfl 32624 ismtybndlem 32775 kelac1 36651 gicabl 36687 eupthvdres 41403 |
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