Metamath Proof Explorer


Theorem f1dom2g

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015) (Proof shortened by BTernaryTau, 25-Sep-2024)

Ref Expression
Assertion f1dom2g
|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )

Proof

Step Hyp Ref Expression
1 f1f
 |-  ( F : A -1-1-> B -> F : A --> B )
2 fex2
 |-  ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3 1 2 syl3an1
 |-  ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F e. _V )
4 3 3coml
 |-  ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F e. _V )
5 f1dom3g
 |-  ( ( F e. _V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )
6 4 5 syld3an1
 |-  ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )