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Theorem resiima 6094
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 5698 . . 3 (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
21a1i 11 . 2 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3 resabs1 6024 . . 3 (𝐵𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
43rneqd 5949 . 2 (𝐵𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5 rnresi 6093 . . 3 ran ( I ↾ 𝐵) = 𝐵
65a1i 11 . 2 (𝐵𝐴 → ran ( I ↾ 𝐵) = 𝐵)
72, 4, 63eqtrd 2781 1 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wss 3951   I cid 5577  ran crn 5686  cres 5687  cima 5688
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  fipreima  9398  psgnunilem1  19511  islinds2  21833  lindsind2  21839  ssidcn  23263  idqtop  23714  fmid  23968  ellspds  33396  rrhre  34022  sitmcl  34353  bj-imdirid  37187  bj-iminvid  37196  poimirlem15  37642  grimidvtxedg  47876  ushggricedg  47896
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