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Theorem rescom 5242
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 3650 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
21reseq2i 5214 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  |`  ( C  i^i  B ) )
3 resres 5230 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
4 resres 5230 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
52, 3, 43eqtr4i 2493 1  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    i^i cin 3434    |` cres 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-opab 4458  df-xp 4953  df-rel 4954  df-res 4959
This theorem is referenced by:  resabs2  5247  setscom  14321  dvres3a  21521  cpnres  21543  dvmptres3  21562
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