Theorem fodomg

Description: An onto function implies dominance of domain over range. Lemma 10.20 of Kunen p. 30. This theorem uses the axiom of choice ac7g . The axiom of choice is not needed for finite sets, see fodomfi . See also fodomnum . (Contributed by NM, 23-Jul-2004) (Proof shortened by BJ, 20-May-2024)

	Ref	Expression
Assertion	fodomg	|- ( A e. V -> ( F : A -onto-> B -> B ~<_ A ) )

Proof

Step	Hyp	Ref	Expression
1		numth3	|- ( A e. V -> A e. dom card )
2		fodomnum	|- ( A e. dom card -> ( F : A -onto-> B -> B ~<_ A ) )
3	1 2	syl	|- ( A e. V -> ( F : A -onto-> B -> B ~<_ A ) )