Metamath Proof Explorer


Theorem symrefref3

Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 . (Contributed by Peter Mazsa, 23-Aug-2021) (Proof modification is discouraged.)

Ref Expression
Assertion symrefref3 x y x R y y R x x dom R y ran R x = y x R y x dom R x R x

Proof

Step Hyp Ref Expression
1 symrefref2 R -1 R I dom R × ran R R I dom R R
2 cnvsym R -1 R x y x R y y R x
3 idinxpss I dom R × ran R R x dom R y ran R x = y x R y
4 idrefALT I dom R R x dom R x R x
5 3 4 bibi12i I dom R × ran R R I dom R R x dom R y ran R x = y x R y x dom R x R x
6 1 2 5 3imtr3i x y x R y y R x x dom R y ran R x = y x R y x dom R x R x