Description: Dominance relation. This variation of brdomg does not require the Axiom of Union. (Contributed by BTernaryTau, 29-Nov-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | brdom2g | |- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 | |- ( x = A -> ( f : x -1-1-> y <-> f : A -1-1-> y ) ) |
|
2 | 1 | exbidv | |- ( x = A -> ( E. f f : x -1-1-> y <-> E. f f : A -1-1-> y ) ) |
3 | f1eq3 | |- ( y = B -> ( f : A -1-1-> y <-> f : A -1-1-> B ) ) |
|
4 | 3 | exbidv | |- ( y = B -> ( E. f f : A -1-1-> y <-> E. f f : A -1-1-> B ) ) |
5 | df-dom | |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y } |
|
6 | 2 4 5 | brabg | |- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) |