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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 5647 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | fnresdm 6621 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
3 | 2 | rneqd 5894 | . 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹) |
4 | 1, 3 | eqtrid 2785 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ran crn 5635 ↾ cres 5636 “ cima 5637 Fn wfn 6492 |
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 2704 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 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 |
This theorem is referenced by: infdifsn 9598 cardinfima 10038 alephfp 10049 dprdf1o 19816 dprd2db 19827 lmhmrnlss 20526 frlmlbs 21219 frlmup3 21222 ellspd 21224 mpfsubrg 21529 pf1subrg 21730 tgrest 22526 uniiccdif 24958 uniioombllem3 24965 dvgt0lem2 25383 f1rnen 31589 cycpmco2rn 32023 fedgmul 32383 zarclsint 32510 eulerpartlemn 33038 matunitlindflem2 36121 poimirlem15 36139 k0004lem1 42507 imasetpreimafvbijlemf 45679 fundcmpsurbijinjpreimafv 45685 |
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