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 ∪ 𝐶 ) ) = ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xpundir	⊢ ( ( 𝐵 ∪ 𝐶 ) × V ) = ( ( 𝐵 × V ) ∪ ( 𝐶 × V ) )
2	1	ineq2i	⊢ ( 𝐴 ∩ ( ( 𝐵 ∪ 𝐶 ) × V ) ) = ( 𝐴 ∩ ( ( 𝐵 × V ) ∪ ( 𝐶 × V ) ) )
3		indi	⊢ ( 𝐴 ∩ ( ( 𝐵 × V ) ∪ ( 𝐶 × V ) ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∪ ( 𝐴 ∩ ( 𝐶 × V ) ) )
4	2 3	eqtri	⊢ ( 𝐴 ∩ ( ( 𝐵 ∪ 𝐶 ) × V ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∪ ( 𝐴 ∩ ( 𝐶 × V ) ) )
5		df-res	⊢ ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( 𝐴 ∩ ( ( 𝐵 ∪ 𝐶 ) × V ) )
6		df-res	⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) )
7		df-res	⊢ ( 𝐴 ↾ 𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) )
8	6 7	uneq12i	⊢ ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∪ ( 𝐴 ∩ ( 𝐶 × V ) ) )
9	4 5 8	3eqtr4i	⊢ ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )