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Theorem imadmres 5846
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 5844 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
21rneqi 5555 . 2 ran (𝐴 ↾ dom (𝐴𝐵)) = ran (𝐴𝐵)
3 df-ima 5325 . 2 (𝐴 “ dom (𝐴𝐵)) = ran (𝐴 ↾ dom (𝐴𝐵))
4 df-ima 5325 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2831 1 (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  dom cdm 5312  ran crn 5313  cres 5314  cima 5315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325
This theorem is referenced by:  ssimaex  6488  fnwelem  7529  imafi  8501  r0weon  9121  limsupgle  14549  kqdisj  21864
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