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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 5051 | . 2 ⊢ ( I “ 𝐴) = ran ( I ↾ 𝐴) | |
2 | imai 5397 | . 2 ⊢ ( I “ 𝐴) = 𝐴 | |
3 | 1, 2 | eqtr3i 2634 | 1 ⊢ ran ( I ↾ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 I cid 4948 ran crn 5039 ↾ cres 5040 “ cima 5041 |
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-eu 2462 df-mo 2463 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-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: resiima 5399 iordsmo 7341 dfac9 8841 relexprng 13634 relexpfld 13637 restid2 15914 sylow1lem2 17837 sylow3lem1 17865 lsslinds 19989 wilthlem3 24596 ausisusgra 25884 cusgraexi 25997 idssxp 28811 diophrw 36340 lnrfg 36708 rclexi 36941 rtrclex 36943 rtrclexi 36947 cnvrcl0 36951 dfrtrcl5 36955 dfrcl2 36985 brfvrcld2 37003 iunrelexp0 37013 relexpiidm 37015 relexp01min 37024 idhe 37101 dvsid 37552 fourierdlem60 39059 fourierdlem61 39060 ausgrusgrb 40395 umgrres1lem 40529 umgrres1 40533 nbupgrres 40592 cusgrexi 40662 cusgrsize 40670 |
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