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Theorem rnresi 6027
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 5646 . 2 ( I “ 𝐴) = ran ( I ↾ 𝐴)
2 imai 6026 . 2 ( I “ 𝐴) = 𝐴
31, 2eqtr3i 2766 1 ran ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   I cid 5530  ran crn 5634  cres 5635  cima 5636
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646
This theorem is referenced by:  resiima  6028  iordsmo  8302  dfac9  10071  relexprng  14930  relexpfld  14933  restid2  17311  sylow1lem2  19379  sylow3lem1  19407  lsslinds  21235  wilthlem3  26417  ausgrusgrb  28063  umgrres1lem  28205  umgrres1  28209  nbupgrres  28259  cusgrexilem2  28337  cusgrsize  28349  cycpmconjslem2  31948  diophrw  41059  lnrfg  41423  rclexi  41868  cnvrcl0  41878  dfrtrcl5  41882  dfrcl2  41927  brfvrcld2  41945  iunrelexp0  41955  relexpiidm  41957  relexp01min  41966  dvsid  42592  fourierdlem60  44378  fourierdlem61  44379  uspgrsprfo  46021
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