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Theorem rnresi 6031
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 5650 . 2 ( I “ 𝐴) = ran ( I ↾ 𝐴)
2 imai 6030 . 2 ( I “ 𝐴) = 𝐴
31, 2eqtr3i 2763 1 ran ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   I cid 5534  ran crn 5638  cres 5639  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  resiima  6032  iordsmo  8307  dfac9  10080  relexprng  14940  relexpfld  14943  restid2  17320  sylow1lem2  19389  sylow3lem1  19417  lsslinds  21260  wilthlem3  26442  ausgrusgrb  28165  umgrres1lem  28307  umgrres1  28311  nbupgrres  28361  cusgrexilem2  28439  cusgrsize  28451  cycpmconjslem2  32060  diophrw  41129  lnrfg  41493  rclexi  41979  cnvrcl0  41989  dfrtrcl5  41993  dfrcl2  42038  brfvrcld2  42056  iunrelexp0  42066  relexpiidm  42068  relexp01min  42077  dvsid  42703  fourierdlem60  44497  fourierdlem61  44498  uspgrsprfo  46140
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