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

Step	Hyp	Ref	Expression
1		incom 3268	. . 3 |- ( B i^i C ) = ( C i^i B )
2	1	reseq2i 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 ) )
5	2, 3, 4	3eqtr4i 2170	1 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )

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  11999