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Definition df-trrel 34869
 Description: Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 34868) is equivalent to satisfying the transitive relation predicate, cf. eltrrelsrel 34876. Alternate definitions are dftrrel2 34872 and dftrrel3 34873. (Contributed by Peter Mazsa, 17-Jul-2021.)
Assertion
Ref Expression
df-trrel ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-trrel
StepHypRef Expression
1 cR . . 3 class 𝑅
21wtrrel 34540 . 2 wff TrRel 𝑅
31cdm 5343 . . . . . . 7 class dom 𝑅
41crn 5344 . . . . . . 7 class ran 𝑅
53, 4cxp 5341 . . . . . 6 class (dom 𝑅 × ran 𝑅)
61, 5cin 3798 . . . . 5 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
76, 6ccom 5347 . . . 4 class ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
87, 6wss 3799 . . 3 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅))
91wrel 5348 . . 3 wff Rel 𝑅
108, 9wa 386 . 2 wff (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
112, 10wb 198 1 wff ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 Colors of variables: wff setvar class This definition is referenced by:  dftrrel2  34872
 Copyright terms: Public domain W3C validator