Metamath Proof Explorer


Theorem f1dom3g

Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024)

Ref Expression
Assertion f1dom3g F V B W F : A 1-1 B A B

Proof

Step Hyp Ref Expression
1 f1eq1 f = F f : A 1-1 B F : A 1-1 B
2 1 spcegv F V F : A 1-1 B f f : A 1-1 B
3 2 imp F V F : A 1-1 B f f : A 1-1 B
4 3 3adant2 F V B W F : A 1-1 B f f : A 1-1 B
5 brdomg B W A B f f : A 1-1 B
6 5 3ad2ant2 F V B W F : A 1-1 B A B f f : A 1-1 B
7 4 6 mpbird F V B W F : A 1-1 B A B