Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | relcnveq | |- ( Rel R -> ( `' R C_ R <-> `' R = R ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnveq3 | |- ( Rel R -> ( R = `' R <-> A. x A. y ( x R y -> y R x ) ) ) |
|
2 | cnvsym | |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) |
|
3 | 1 2 | bitr4di | |- ( Rel R -> ( R = `' R <-> `' R C_ R ) ) |
4 | eqcom | |- ( R = `' R <-> `' R = R ) |
|
5 | 3 4 | bitr3di | |- ( Rel R -> ( `' R C_ R <-> `' R = R ) ) |