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Theorem resima 6035
Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.)
Assertion
Ref Expression
resima ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)

Proof of Theorem resima
StepHypRef Expression
1 residm 6030 . . 3 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
21rneqi 5951 . 2 ran ((𝐴𝐵) ↾ 𝐵) = ran (𝐴𝐵)
3 df-ima 5702 . 2 ((𝐴𝐵) “ 𝐵) = ran ((𝐴𝐵) ↾ 𝐵)
4 df-ima 5702 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2773 1 ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  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-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:  isarep2  6659  f1imacnv  6865  foimacnv  6866  dffv2  7004  fssrescdmd  7146  islindf4  21876  qtopres  23722  aks6d1c6lem4  42155
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