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 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 5543 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
21rneqi 5273 . 2 ran (𝐴 ↾ dom (𝐴𝐵)) = ran (𝐴𝐵)
3 df-ima 5051 . 2 (𝐴 “ dom (𝐴𝐵)) = ran (𝐴 ↾ dom (𝐴𝐵))
4 df-ima 5051 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2642 1 (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  ssimaex  6173  fnwelem  7179  imafi  8142  r0weon  8718  limsupgle  14056  kqdisj  21345
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