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Theorem rnresi 5589
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 5231 . 2 ( I “ 𝐴) = ran ( I ↾ 𝐴)
2 imai 5588 . 2 ( I “ 𝐴) = 𝐴
31, 2eqtr3i 2748 1 ran ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1596   I cid 5127  ran crn 5219  cres 5220  cima 5221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231
This theorem is referenced by:  resiima  5590  idssxp  6122  iordsmo  7574  dfac9  9071  relexprng  13906  relexpfld  13909  restid2  16214  sylow1lem2  18135  sylow3lem1  18163  lsslinds  20293  wilthlem3  24916  ausgrusgrb  26180  umgrres1lem  26322  umgrres1  26326  nbupgrres  26385  cusgrexilem2  26469  cusgrsize  26481  diophrw  37741  lnrfg  38108  rclexi  38341  rtrclex  38343  rtrclexi  38347  cnvrcl0  38351  dfrtrcl5  38355  dfrcl2  38385  brfvrcld2  38403  iunrelexp0  38413  relexpiidm  38415  relexp01min  38424  idhe  38500  dvsid  38949  fourierdlem60  40803  fourierdlem61  40804  uspgrsprfo  42183
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