Theorem resundi

Description: Distributive law for restriction over union. Theorem 31 of Suppes p. 65. (Contributed by NM, 30-Sep-2002)

	Ref	Expression
Assertion	resundi	|- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Proof

Step	Hyp	Ref	Expression
1		xpundir	|- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i	|- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi	|- ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
4	2 3	eqtri	|- ( A i^i ( ( B u. C ) X. _V ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
5		df-res	|- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res	|- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res	|- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6 7	uneq12i	|- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4 5 8	3eqtr4i	|- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )