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 ) |
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 ) |