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Metamath Proof Explorer

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 ) )