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Metamath Proof Explorer

Theorem refsymrel2

Description: A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the (I i^i ( dom R X. ran R ) ) C R version of dfrefrel2 , cf. the comment of dfrefrels2 . (Contributed by Peter Mazsa, 23-Aug-2021)

Ref Expression
Assertion refsymrel2 ( ( RefRel 𝑅 ∧ SymRel 𝑅 ) ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 dfrefrel2 ⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
2 dfsymrel2 ⊢ ( SymRel 𝑅 ↔ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )
3 1 2 anbi12i ⊢ ( ( RefRel 𝑅 ∧ SymRel 𝑅 ) ↔ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ∧ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) ) )
4 anandi3r ⊢ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ↔ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ∧ ( ◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) ) )
5 3anan32 ⊢ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ↔ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) )
6 3 4 5 3bitr2i ⊢ ( ( RefRel 𝑅 ∧ SymRel 𝑅 ) ↔ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) )
7 symrefref2 ⊢ ( ◡ 𝑅 ⊆ 𝑅 → ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅 ) ⊆ 𝑅 ) )
8 7 pm5.32ri ⊢ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ↔ ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) )
9 8 anbi1i ⊢ ( ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) )
10 6 9 bitri ⊢ ( ( RefRel 𝑅 ∧ SymRel 𝑅 ) ↔ ( ( ( I ↾ dom 𝑅 ) ⊆ 𝑅 ∧ ◡ 𝑅 ⊆ 𝑅 ) ∧ Rel 𝑅 ) )