Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5051 | . 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | fnresdm 5914 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
3 | 2 | rneqd 5274 | . 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹) |
4 | 1, 3 | syl5eq 2656 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ran crn 5039 ↾ cres 5040 “ cima 5041 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 |
This theorem is referenced by: infdifsn 8437 carduniima 8802 cardinfima 8803 alephfp 8814 dprdf1o 18254 dprd2db 18265 lmhmrnlss 18871 mpfsubrg 19353 pf1subrg 19533 frlmlbs 19955 frlmup3 19958 ellspd 19960 tgrest 20773 uniiccdif 23152 uniioombllem3 23159 dvgt0lem2 23570 eulerpartlemn 29770 matunitlindflem2 32576 poimirlem15 32594 k0004lem1 37465 |
Copyright terms: Public domain | W3C validator |