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Theorem imadmres 6253
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 6251 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
21rneqi 5947 . 2 ran (𝐴 ↾ dom (𝐴𝐵)) = ran (𝐴𝐵)
3 df-ima 5697 . 2 (𝐴 “ dom (𝐴𝐵)) = ran (𝐴 ↾ dom (𝐴𝐵))
4 df-ima 5697 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2774 1 (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  dom cdm 5684  ran crn 5685  cres 5686  cima 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  ssimaex  6993  fnwelem  8157  imafi  9354  r0weon  10053  limsupgle  15514  kqdisj  23741  isubgruhgr  47859
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