Description: One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | laut1o.b | |- B = ( Base ` K ) |
|
laut1o.i | |- I = ( LAut ` K ) |
||
Assertion | laut11 | |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | laut1o.b | |- B = ( Base ` K ) |
|
2 | laut1o.i | |- I = ( LAut ` K ) |
|
3 | 1 2 | laut1o | |- ( ( K e. V /\ F e. I ) -> F : B -1-1-onto-> B ) |
4 | f1of1 | |- ( F : B -1-1-onto-> B -> F : B -1-1-> B ) |
|
5 | 3 4 | syl | |- ( ( K e. V /\ F e. I ) -> F : B -1-1-> B ) |
6 | f1fveq | |- ( ( F : B -1-1-> B /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) ) |
|
7 | 5 6 | sylan | |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) ) |