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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-trrel | Structured version Visualization version GIF version |
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 37064) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 37072. Alternate definitions are dftrrel2 37068 and dftrrel3 37069. (Contributed by Peter Mazsa, 17-Jul-2021.) |
Ref | Expression |
---|---|
df-trrel | ⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cR | . . 3 class 𝑅 | |
2 | 1 | wtrrel 36678 | . 2 wff TrRel 𝑅 |
3 | 1 | cdm 5638 | . . . . . . 7 class dom 𝑅 |
4 | 1 | crn 5639 | . . . . . . 7 class ran 𝑅 |
5 | 3, 4 | cxp 5636 | . . . . . 6 class (dom 𝑅 × ran 𝑅) |
6 | 1, 5 | cin 3914 | . . . . 5 class (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
7 | 6, 6 | ccom 5642 | . . . 4 class ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) |
8 | 7, 6 | wss 3915 | . . 3 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
9 | 1 | wrel 5643 | . . 3 wff Rel 𝑅 |
10 | 8, 9 | wa 397 | . 2 wff (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) |
11 | 2, 10 | wb 205 | 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 37068 |
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