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Theorem rescom 5173
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 3535 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
21reseq2i 5145 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  |`  ( C  i^i  B ) )
3 resres 5161 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
4 resres 5161 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
52, 3, 43eqtr4i 2468 1  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    i^i cin 3321    |` cres 4882
This theorem is referenced by:  resabs2  5178  setscom  13499  dvres3a  19803  cpnres  19825  dvmptres3  19844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887  df-res 4892
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