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Theorem rnresss 6018
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 6003 . 2 (𝐴𝐵) ⊆ 𝐴
21rnssi 5938 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3946  ran crn 5675  cres 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686
This theorem is referenced by:  gsumhashmul  32929  nelrnres  44830  limsupvaluz2  45395  supcnvlimsup  45397  limsupgtlem  45434  sge0split  46066
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