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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 5571 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | fnresdm 6469 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
3 | 2 | rneqd 5811 | . 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹) |
4 | 1, 3 | syl5eq 2871 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ran crn 5559 ↾ cres 5560 “ cima 5561 Fn wfn 6353 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-fun 6360 df-fn 6361 |
This theorem is referenced by: infdifsn 9123 cardinfima 9526 alephfp 9537 dprdf1o 19157 dprd2db 19168 lmhmrnlss 19825 mpfsubrg 20319 pf1subrg 20514 frlmlbs 20944 frlmup3 20947 ellspd 20949 tgrest 21770 uniiccdif 24182 uniioombllem3 24189 dvgt0lem2 24603 f1rnen 30377 cycpmco2rn 30771 fedgmul 31031 eulerpartlemn 31643 matunitlindflem2 34893 poimirlem15 34911 k0004lem1 40503 imasetpreimafvbijlemf 43568 fundcmpsurbijinjpreimafv 43574 |
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