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Theorem resiima 6032
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 5650 . . 3 (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
21a1i 11 . 2 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3 resabs1 5971 . . 3 (𝐵𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
43rneqd 5897 . 2 (𝐵𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5 rnresi 6031 . . 3 ran ( I ↾ 𝐵) = 𝐵
65a1i 11 . 2 (𝐵𝐴 → ran ( I ↾ 𝐵) = 𝐵)
72, 4, 63eqtrd 2777 1 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3914   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-10 2138  ax-12 2172  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-nf 1787  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:  fipreima  9308  psgnunilem1  19283  islinds2  21242  lindsind2  21248  ssidcn  22629  idqtop  23080  fmid  23334  ellspds  32211  rrhre  32666  sitmcl  33015  bj-imdirid  35707  bj-iminvid  35716  poimirlem15  36143  isomgreqve  46107  ushrisomgr  46123
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