Theorem rnresi 6075

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

Step	Hyp	Ref	Expression
1		df-ima 5690	. 2 ⊢ ( I “ 𝐴) = ran ( I ↾ 𝐴)
2		imai 6074	. 2 ⊢ ( I “ 𝐴) = 𝐴
3	1, 2	eqtr3i 2760	1 ⊢ ran ( I ↾ 𝐴) = 𝐴

Syntax hints:   = wceq 1539   I cid 5574  ran crn 5678   ↾ cres 5679   “ cima 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690

This theorem is referenced by:  resiima  6076  iordsmo  8361  dfac9  10135  relexprng  14999  relexpfld  15002  restid2  17382  sylow1lem2  19510  sylow3lem1  19538  lsslinds  21607  wilthlem3  26808  ausgrusgrb  28690  umgrres1lem  28832  umgrres1  28836  nbupgrres  28886  cusgrexilem2  28964  cusgrsize  28976  cycpmconjslem2  32582  diophrw  41801  lnrfg  42165  rclexi  42670  cnvrcl0  42680  dfrtrcl5  42684  dfrcl2  42729  brfvrcld2  42747  iunrelexp0  42757  relexpiidm  42759  relexp01min  42768  dvsid  43394  fourierdlem60  45182  fourierdlem61  45183  uspgrsprfo  46826