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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 6024 | . 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | fof 6757 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6680 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
4 | 3 | imaeq2d 6014 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
5 | forn 6760 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | 1, 4, 5 | 3eqtr3a 2801 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 dom cdm 5634 ran crn 5635 “ cima 5637 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: foimacnv 6802 domunfican 9265 fiint 9269 fodomfi 9270 cantnflt2 9610 cantnfp1lem3 9617 enfin1ai 10321 symgfixelsi 19218 dprdf1o 19812 lmimlbs 21245 cncmp 22746 cmpfi 22762 cnconn 22776 qtopval2 23050 elfm3 23304 rnelfm 23307 fmfnfmlem2 23309 fmfnfm 23312 eupthvdres 29182 pjordi 31118 qtophaus 32420 poimirlem1 36082 poimirlem2 36083 poimirlem3 36084 poimirlem4 36085 poimirlem5 36086 poimirlem6 36087 poimirlem7 36088 poimirlem9 36090 poimirlem10 36091 poimirlem11 36092 poimirlem12 36093 poimirlem14 36095 poimirlem16 36097 poimirlem17 36098 poimirlem19 36100 poimirlem20 36101 poimirlem22 36103 poimirlem23 36104 poimirlem24 36105 poimirlem25 36106 poimirlem29 36110 poimirlem31 36112 ovoliunnfl 36123 voliunnfl 36125 volsupnfl 36126 ismtybndlem 36268 riccrng1 40706 ricdrng1 40720 kelac1 41393 gicabl 41429 |
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