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Theorem rescom 4844
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 3268 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
21reseq2i 4816 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  |`  ( C  i^i  B ) )
3 resres 4831 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
4 resres 4831 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
52, 3, 43eqtr4i 2170 1  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    i^i cin 3070    |` cres 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546  df-res 4551
This theorem is referenced by:  resabs2  4850  setscom  12009
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