| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > rnresi | Structured version Visualization version GIF version | ||
| Description: The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.) |
| Ref | Expression |
|---|---|
| rnresi | ⊢ ran ( I ↾ 𝐴) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ima 5698 | . 2 ⊢ ( I “ 𝐴) = ran ( I ↾ 𝐴) | |
| 2 | imai 6092 | . 2 ⊢ ( I “ 𝐴) = 𝐴 | |
| 3 | 1, 2 | eqtr3i 2767 | 1 ⊢ ran ( I ↾ 𝐴) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 I cid 5577 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: resiima 6094 iordsmo 8397 dfac9 10177 relexprng 15085 relexpfld 15088 restid2 17475 sylow1lem2 19617 sylow3lem1 19645 lsslinds 21851 wilthlem3 27113 ausgrusgrb 29182 umgrres1lem 29327 umgrres1 29331 nbupgrres 29381 cusgrexilem2 29459 cusgrsize 29472 cycpmconjslem2 33175 diophrw 42770 lnrfg 43131 rclexi 43628 cnvrcl0 43638 dfrtrcl5 43642 dfrcl2 43687 brfvrcld2 43705 iunrelexp0 43715 relexpiidm 43717 relexp01min 43726 dvsid 44350 fourierdlem60 46181 fourierdlem61 46182 stgredg 47923 gpgedg 48004 uspgrsprfo 48064 |
| Copyright terms: Public domain | W3C validator |