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Metamath Proof Explorer

Theorem dftrrel2

Description: Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021)

Ref Expression
Assertion dftrrel2 ( TrRel 𝑅 ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ Rel 𝑅 ) )

Proof

Step Hyp Ref Expression
1 df-trrel ⊢ ( TrRel 𝑅 ↔ ( ( ( 𝑅 ∩ ( dom 𝑅 × ran 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 3 coeq12d ⊢ ( Rel 𝑅 → ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∘ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ) = ( 𝑅 ∘ 𝑅 ) )
5 4 3 sseq12d ⊢ ( Rel 𝑅 → ( ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∘ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ) )
6 5 pm5.32ri ⊢ ( ( ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∘ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
7 1 6 bitri ⊢ ( TrRel 𝑅 ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ Rel 𝑅 ) )