Theorem coexg

Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998)

	Ref	Expression
Assertion	coexg	|- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )

Proof

Step	Hyp	Ref	Expression
1		cossxp	|- ( A o. B ) C_ ( dom B X. ran A )
2		dmexg	|- ( B e. W -> dom B e. _V )
3		rnexg	|- ( A e. V -> ran A e. _V )
4		xpexg	|- ( ( dom B e. _V /\ ran A e. _V ) -> ( dom B X. ran A ) e. _V )
5	2 3 4	syl2anr	|- ( ( A e. V /\ B e. W ) -> ( dom B X. ran A ) e. _V )
6		ssexg	|- ( ( ( A o. B ) C_ ( dom B X. ran A ) /\ ( dom B X. ran A ) e. _V ) -> ( A o. B ) e. _V )
7	1 5 6	sylancr	|- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )