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Metamath Proof Explorer

Theorem dfrefrel2

Description: Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021)

Ref Expression
Assertion dfrefrel2 ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 df-refrel ⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( 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 sseq2d ⊢ ( Rel 𝑅 → ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ) )
5 4 pm5.32ri ⊢ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
6 1 5 bitri ⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )