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Theorem rescom 5146
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 3656 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
21reseq2i 5119 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  |`  ( C  i^i  B ) )
3 resres 5134 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
4 resres 5134 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
52, 3, 43eqtr4i 2462 1  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    i^i cin 3436    |` cres 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-opab 4481  df-xp 4857  df-rel 4858  df-res 4863
This theorem is referenced by:  resabs2  5152  setscom  15146  dvres3a  22861  cpnres  22883  dvmptres3  22902
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