MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnresss Structured version   Visualization version   GIF version

Theorem rnresss 6034
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 6018 . 2 (𝐴𝐵) ⊆ 𝐴
21rnssi 5950 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3950  ran crn 5685  cres 5686
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
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-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-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696
This theorem is referenced by:  imadifssran  6170  gsumhashmul  33065  nelrnres  45197  limsupvaluz2  45758  supcnvlimsup  45760  limsupgtlem  45797  sge0split  46429
  Copyright terms: Public domain W3C validator