Theorem dmin

Description: The domain of an intersection is included in the intersection of the domains. Theorem 6 of Suppes p. 60. (Contributed by NM, 15-Sep-2004)

	Ref	Expression
Assertion	dmin	|- dom ( A i^i B ) C_ ( dom A i^i dom B )

Proof

Step	Hyp	Ref	Expression
1		19.40	|- ( E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) -> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
2		vex	|- x e. _V
3	2	eldm2	|- ( x e. dom ( A i^i B ) <-> E. y <. x , y >. e. ( A i^i B ) )
4		elin	|- ( <. x , y >. e. ( A i^i B ) <-> ( <. x , y >. e. A /\ <. x , y >. e. B ) )
5	4	exbii	|- ( E. y <. x , y >. e. ( A i^i B ) <-> E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) )
6	3 5	bitri	|- ( x e. dom ( A i^i B ) <-> E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) )
7		elin	|- ( x e. ( dom A i^i dom B ) <-> ( x e. dom A /\ x e. dom B ) )
8	2	eldm2	|- ( x e. dom A <-> E. y <. x , y >. e. A )
9	2	eldm2	|- ( x e. dom B <-> E. y <. x , y >. e. B )
10	8 9	anbi12i	|- ( ( x e. dom A /\ x e. dom B ) <-> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
11	7 10	bitri	|- ( x e. ( dom A i^i dom B ) <-> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
12	1 6 11	3imtr4i	|- ( x e. dom ( A i^i B ) -> x e. ( dom A i^i dom B ) )
13	12	ssriv	|- dom ( A i^i B ) C_ ( dom A i^i dom B )