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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 5701 | . 2 ⊢ ( I “ 𝐴) = ran ( I ↾ 𝐴) | |
2 | imai 6093 | . 2 ⊢ ( I “ 𝐴) = 𝐴 | |
3 | 1, 2 | eqtr3i 2764 | 1 ⊢ ran ( I ↾ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 I cid 5581 ran crn 5689 ↾ cres 5690 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: resiima 6095 iordsmo 8395 dfac9 10174 relexprng 15081 relexpfld 15084 restid2 17476 sylow1lem2 19631 sylow3lem1 19659 lsslinds 21868 wilthlem3 27127 ausgrusgrb 29196 umgrres1lem 29341 umgrres1 29345 nbupgrres 29395 cusgrexilem2 29473 cusgrsize 29486 cycpmconjslem2 33157 diophrw 42746 lnrfg 43107 rclexi 43604 cnvrcl0 43614 dfrtrcl5 43618 dfrcl2 43663 brfvrcld2 43681 iunrelexp0 43691 relexpiidm 43693 relexp01min 43702 dvsid 44326 fourierdlem60 46121 fourierdlem61 46122 stgredg 47858 gpgedg 47939 uspgrsprfo 47991 |
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