Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnresi Structured version   Visualization version   GIF version

Theorem rnresi 5398
 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 5051 . 2 ( I “ 𝐴) = ran ( I ↾ 𝐴)
2 imai 5397 . 2 ( I “ 𝐴) = 𝐴
31, 2eqtr3i 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
 Copyright terms: Public domain W3C validator