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 AVBWF:A1-1BAB

Proof

Step Hyp Ref Expression
1 f1f F:A1-1BF:AB
2 fex2 F:ABAVBWFV
3 1 2 syl3an1 F:A1-1BAVBWFV
4 3 3coml AVBWF:A1-1BFV
5 f1dom3g FVBWF:A1-1BAB
6 4 5 syld3an1 AVBWF:A1-1BAB