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Theorem resiima 4626
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 4301 . . 3 (( I ↾ A) “ B) = ran (( I ↾ A) ↾ B)
21a1i 9 . 2 (BA → (( I ↾ A) “ B) = ran (( I ↾ A) ↾ B))
3 resabs1 4583 . . 3 (BA → (( I ↾ A) ↾ B) = ( I ↾ B))
43rneqd 4506 . 2 (BA → ran (( I ↾ A) ↾ B) = ran ( I ↾ B))
5 rnresi 4625 . . 3 ran ( I ↾ B) = B
65a1i 9 . 2 (BA → ran ( I ↾ B) = B)
72, 4, 63eqtrd 2073 1 (BA → (( I ↾ A) “ B) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wss 2911   I cid 4016  ran crn 4289  cres 4290  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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