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Theorem rnresss 38857
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 5383 . 2 (𝐴𝐵) ⊆ 𝐴
2 rnss 5316 . 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 3556  ran crn 5077  cres 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088
This theorem is referenced by:  nelrnres  38866  limsupvaluz2  39392  supcnvlimsup  39394  sge0split  39949
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