This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )