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Theorem rnresi 6032
Description: The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
rnresi ran ( I ↾ 𝐴) = 𝐴

Proof of Theorem rnresi
StepHypRef Expression
1 df-ima 5651 . 2 ( I “ 𝐴) = ran ( I ↾ 𝐴)
2 imai 6031 . 2 ( I “ 𝐴) = 𝐴
31, 2eqtr3i 2761 1 ran ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   I cid 5535  ran crn 5639  cres 5640  cima 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  resiima  6033  iordsmo  8308  dfac9  10081  relexprng  14943  relexpfld  14946  restid2  17326  sylow1lem2  19395  sylow3lem1  19423  lsslinds  21274  wilthlem3  26456  ausgrusgrb  28179  umgrres1lem  28321  umgrres1  28325  nbupgrres  28375  cusgrexilem2  28453  cusgrsize  28465  cycpmconjslem2  32074  diophrw  41140  lnrfg  41504  rclexi  42009  cnvrcl0  42019  dfrtrcl5  42023  dfrcl2  42068  brfvrcld2  42086  iunrelexp0  42096  relexpiidm  42098  relexp01min  42107  dvsid  42733  fourierdlem60  44527  fourierdlem61  44528  uspgrsprfo  46170
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