Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Peter Mazsa
Equivalence relations
eqvrelsym
Metamath Proof Explorer
Description: An equivalence relation is symmetric. (Contributed by NM , 4-Jun-1995)
(Revised by Mario Carneiro , 12-Aug-2015) (Revised by Peter Mazsa , 2-Jun-2019)
Ref
Expression
Hypotheses
eqvrelsym.1
⊢ ( 𝜑 → EqvRel 𝑅 )
eqvrelsym.2
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
Assertion
eqvrelsym
⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )
Proof
Step
Hyp
Ref
Expression
1
eqvrelsym.1
⊢ ( 𝜑 → EqvRel 𝑅 )
2
eqvrelsym.2
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3
eqvrelrel
⊢ ( EqvRel 𝑅 → Rel 𝑅 )
4
relbrcnvg
⊢ ( Rel 𝑅 → ( 𝐵 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) )
5
1 3 4
3syl
⊢ ( 𝜑 → ( 𝐵 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) )
6
2 5
mpbird
⊢ ( 𝜑 → 𝐵 ◡ 𝑅 𝐴 )
7
eqvrelsymrel
⊢ ( EqvRel 𝑅 → SymRel 𝑅 )
8
dfsymrel2
⊢ ( SymRel 𝑅 ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )
9
8
simplbi
⊢ ( SymRel 𝑅 → ◡ 𝑅 ⊆ 𝑅 )
10
1 7 9
3syl
⊢ ( 𝜑 → ◡ 𝑅 ⊆ 𝑅 )
11
10
ssbrd
⊢ ( 𝜑 → ( 𝐵 ◡ 𝑅 𝐴 → 𝐵 𝑅 𝐴 ) )
12
6 11
mpd
⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )