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Theorem rnresss 41730
Description: The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
rnresss ran (𝐴𝐵) ⊆ ran 𝐴

Proof of Theorem rnresss
StepHypRef Expression
1 resss 5865 . 2 (𝐴𝐵) ⊆ 𝐴
21rnssi 5797 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3919  ran crn 5543  cres 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554
This theorem is referenced by:  nelrnres  41739  limsupvaluz2  42306  supcnvlimsup  42308  limsupgtlem  42345  sge0split  42974
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