Theorem dfsymrel2

Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019) (Revised by Peter Mazsa, 17-Aug-2021)

	Ref	Expression
Assertion	dfsymrel2	⊢ ( SymRel 𝑅 ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		df-symrel	⊢ ( SymRel 𝑅 ↔ ( ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )
2		dfrel6	⊢ ( Rel 𝑅 ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 )
3	2	biimpi	⊢ ( Rel 𝑅 → ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 )
4	3	cnveqd	⊢ ( Rel 𝑅 → ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = ◡ 𝑅 )
5	4 3	sseq12d	⊢ ( Rel 𝑅 → ( ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ ◡ 𝑅 ⊆ 𝑅 ) )
6	5	pm5.32ri	⊢ ( ( ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )
7	1 6	bitri	⊢ ( SymRel 𝑅 ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )