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Theorem rescom 5130
Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
rescom  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )

Proof of Theorem rescom
StepHypRef Expression
1 incom 3538 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
21reseq2i 5102 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  |`  ( C  i^i  B ) )
3 resres 5118 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
4 resres 5118 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
52, 3, 43eqtr4i 2468 1  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    i^i cin 3322    |` cres 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-opab 4346  df-xp 4841  df-rel 4842  df-res 4847
This theorem is referenced by:  resabs2  5135  setscom  14196  dvres3a  21364  cpnres  21386  dvmptres3  21405
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