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Theorem resima 5970
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 5969 . . 3 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
21rneqi 5891 . 2 ran ((𝐴𝐵) ↾ 𝐵) = ran (𝐴𝐵)
3 df-ima 5645 . 2 ((𝐴𝐵) “ 𝐵) = ran ((𝐴𝐵) ↾ 𝐵)
4 df-ima 5645 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2776 1 ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ran crn 5633  cres 5634  cima 5635
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645
This theorem is referenced by:  isarep2  6590  f1imacnv  6798  foimacnv  6799  dffv2  6934  islindf4  21197  qtopres  23001
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