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Theorem resiima 4867
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 4522 . . 3 (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)
21a1i 9 . 2 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵))
3 resabs1 4818 . . 3 (𝐵𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵))
43rneqd 4738 . 2 (𝐵𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵))
5 rnresi 4866 . . 3 ran ( I ↾ 𝐵) = 𝐵
65a1i 9 . 2 (𝐵𝐴 → ran ( I ↾ 𝐵) = 𝐵)
72, 4, 63eqtrd 2154 1 (𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wss 3041   I cid 4180  ran crn 4510  cres 4511  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  ssidcn  12306
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