Theorem dom2

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ) , as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Hypotheses	dom2.1	|- ( x e. A -> C e. B )
	dom2.2	|- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )
Assertion	dom2	|- ( B e. V -> A ~<_ B )

Proof

Step	Hyp	Ref	Expression
1		dom2.1	|- ( x e. A -> C e. B )
2		dom2.2	|- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )
3		eqid	|- A = A
4	1	a1i	|- ( A = A -> ( x e. A -> C e. B ) )
5	2	a1i	|- ( A = A -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
6	4 5	dom2d	|- ( A = A -> ( B e. V -> A ~<_ B ) )
7	3 6	ax-mp	|- ( B e. V -> A ~<_ B )