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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 5698 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
| 2 | fnresdm 6687 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
| 3 | 2 | rneqd 5949 | . 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹) |
| 4 | 1, 3 | eqtrid 2789 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ran crn 5686 ↾ cres 5687 “ cima 5688 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: infdifsn 9697 cardinfima 10137 alephfp 10148 dprdf1o 20052 dprd2db 20063 rnrhmsubrg 20605 lmhmrnlss 21049 frlmlbs 21817 frlmup3 21820 ellspd 21822 mpfsubrg 22127 pf1subrg 22352 tgrest 23167 uniiccdif 25613 uniioombllem3 25620 dvgt0lem2 26042 f1rnen 32639 cycpmco2rn 33145 r1pquslmic 33631 fedgmul 33682 zarclsint 33871 eulerpartlemn 34383 matunitlindflem2 37624 poimirlem15 37642 aks6d1c6lem3 42173 aks6d1c6lem5 42178 aks6d1c7lem1 42181 k0004lem1 44160 3f1oss1 47087 imasetpreimafvbijlemf 47388 fundcmpsurbijinjpreimafv 47394 |
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