This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )