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 ) )