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Theorem rnresss 37451
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  ( A  |`  B )  C_  ran  A

Proof of Theorem rnresss
StepHypRef Expression
1 resss 5128 . 2  |-  ( A  |`  B )  C_  A
2 rnss 5063 . 2  |-  ( ( A  |`  B )  C_  A  ->  ran  ( A  |`  B )  C_  ran  A )
31, 2ax-mp 5 1  |-  ran  ( A  |`  B )  C_  ran  A
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3404   ran crn 4835    |` cres 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846
This theorem is referenced by:  nelrnres  37462  sge0split  38251
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