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

Theorem rnresi 5169
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 4835 . 2  |-  (  _I  " A )  =  ran  (  _I  |`  A )
2 imai 5168 . 2  |-  (  _I  " A )  =  A
31, 2eqtr3i 2433 1  |-  ran  (  _I  |`  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    _I cid 4732   ran crn 4823    |` cres 4824   "cima 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835
This theorem is referenced by:  resiima  5170  iordsmo  7060  dfac9  8547  relexprng  13026  relexpfld  13029  restid2  15043  sylow1lem2  16941  sylow3lem1  16969  lsslinds  19156  wilthlem3  23723  ausisusgra  24759  cusgraexi  24872  idssxp  27896  diophrw  35033  lnrfg  35412  dfrcl2  35633  brfvrcld2  35651  iunrelexp0  35661  relexpiidm  35663  relexp01min  35672  idhe  35748  dvsid  36064  fourierdlem60  37298  fourierdlem61  37299  usgresvm1  38053  usgresvm1ALT  38057
  Copyright terms: Public domain W3C validator