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Mirrors > Home > MPE Home > Th. List > fnima | Structured version Visualization version GIF version |
Description: The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fnima | ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5562 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | fnresdm 6460 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
3 | 2 | rneqd 5802 | . 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹) |
4 | 1, 3 | syl5eq 2868 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ran crn 5550 ↾ cres 5551 “ cima 5552 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 |
This theorem is referenced by: infdifsn 9109 cardinfima 9512 alephfp 9523 dprdf1o 19085 dprd2db 19096 lmhmrnlss 19753 mpfsubrg 20246 pf1subrg 20441 frlmlbs 20871 frlmup3 20874 ellspd 20876 tgrest 21697 uniiccdif 24108 uniioombllem3 24115 dvgt0lem2 24529 f1rnen 30303 cycpmco2rn 30695 fedgmul 30927 eulerpartlemn 31539 matunitlindflem2 34771 poimirlem15 34789 k0004lem1 40377 |
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