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Theorem imadmres 6256
Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imadmres (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem imadmres
StepHypRef Expression
1 resdmres 6254 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
21rneqi 5951 . 2 ran (𝐴 ↾ dom (𝐴𝐵)) = ran (𝐴𝐵)
3 df-ima 5702 . 2 (𝐴 “ dom (𝐴𝐵)) = ran (𝐴 ↾ dom (𝐴𝐵))
4 df-ima 5702 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2773 1 (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  dom cdm 5689  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:  ssimaex  6994  fnwelem  8155  imafi  9351  r0weon  10050  limsupgle  15510  kqdisj  23756  isubgruhgr  47792
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