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Theorem resima 5976
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 5975 . . 3 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
21rneqi 5897 . 2 ran ((𝐴𝐵) ↾ 𝐵) = ran (𝐴𝐵)
3 df-ima 5651 . 2 ((𝐴𝐵) “ 𝐵) = ran ((𝐴𝐵) ↾ 𝐵)
4 df-ima 5651 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2775 1 ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ran crn 5639  cres 5640  cima 5641
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  isarep2  6597  f1imacnv  6805  foimacnv  6806  dffv2  6941  islindf4  21260  qtopres  23065
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