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Theorem rnresss 40292
 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 5671 . 2 (𝐴𝐵) ⊆ 𝐴
2 rnss 5599 . 2 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
31, 2ax-mp 5 1 ran (𝐴𝐵) ⊆ ran 𝐴
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3792  ran crn 5356   ↾ cres 5357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367 This theorem is referenced by:  nelrnres  40301  limsupvaluz2  40882  supcnvlimsup  40884  limsupgtlem  40921  sge0split  41554
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