Theorem numdom

Description: A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	numdom	|- ( ( A e. dom card /\ B ~<_ A ) -> B e. dom card )

Proof

Step	Hyp	Ref	Expression
1		cardon	|- ( card ` A ) e. On
2		cardid2	|- ( A e. dom card -> ( card ` A ) ~~ A )
3		domen2	|- ( ( card ` A ) ~~ A -> ( B ~<_ ( card ` A ) <-> B ~<_ A ) )
4	2 3	syl	|- ( A e. dom card -> ( B ~<_ ( card ` A ) <-> B ~<_ A ) )
5	4	biimpar	|- ( ( A e. dom card /\ B ~<_ A ) -> B ~<_ ( card ` A ) )
6		ondomen	|- ( ( ( card ` A ) e. On /\ B ~<_ ( card ` A ) ) -> B e. dom card )
7	1 5 6	sylancr	|- ( ( A e. dom card /\ B ~<_ A ) -> B e. dom card )