Theorem rescom

Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998)

	Ref	Expression
Assertion	rescom	|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )

Proof

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