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Theorem resiima 6096
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 5702 . . 3 (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
21a1i 11 . 2 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3 resabs1 6027 . . 3 (𝐵𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
43rneqd 5952 . 2 (𝐵𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5 rnresi 6095 . . 3 ran ( I ↾ 𝐵) = 𝐵
65a1i 11 . 2 (𝐵𝐴 → ran ( I ↾ 𝐵) = 𝐵)
72, 4, 63eqtrd 2779 1 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wss 3963   I cid 5582  ran crn 5690  cres 5691  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  fipreima  9396  psgnunilem1  19526  islinds2  21851  lindsind2  21857  ssidcn  23279  idqtop  23730  fmid  23984  ellspds  33376  rrhre  33984  sitmcl  34333  bj-imdirid  37169  bj-iminvid  37178  poimirlem15  37622  grimidvtxedg  47814  ushggricedg  47834
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