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Theorem resiima 5399
 Description: The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
Assertion
Ref Expression
resiima (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Proof of Theorem resiima
StepHypRef Expression
1 df-ima 5051 . . 3 (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
21a1i 11 . 2 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3 resabs1 5347 . . 3 (𝐵𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
43rneqd 5274 . 2 (𝐵𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5 rnresi 5398 . . 3 ran ( I ↾ 𝐵) = 𝐵
65a1i 11 . 2 (𝐵𝐴 → ran ( I ↾ 𝐵) = 𝐵)
72, 4, 63eqtrd 2648 1 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540   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:  fipreima  8155  psgnunilem1  17736  islinds2  19971  lindsind2  19977  ssidcn  20869  idqtop  21319  fmid  21574  rrhre  29393  sitmcl  29740  poimirlem15  32594
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