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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 5568 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | fnresdm 6466 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
3 | 2 | rneqd 5808 | . 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 1537 ran crn 5556 ↾ cres 5557 “ cima 5558 Fn wfn 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 |
This theorem is referenced by: infdifsn 9120 cardinfima 9523 alephfp 9534 dprdf1o 19154 dprd2db 19165 lmhmrnlss 19822 mpfsubrg 20316 pf1subrg 20511 frlmlbs 20941 frlmup3 20944 ellspd 20946 tgrest 21767 uniiccdif 24179 uniioombllem3 24186 dvgt0lem2 24600 f1rnen 30374 cycpmco2rn 30767 fedgmul 31027 eulerpartlemn 31639 matunitlindflem2 34904 poimirlem15 34922 k0004lem1 40546 imasetpreimafvbijlemf 43610 fundcmpsurbijinjpreimafv 43616 |
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