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Metamath Proof Explorer

Theorem eltrrels3

Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021)

Ref Expression
Assertion eltrrels3 ( 𝑅 ∈ TrRels ↔ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ∧ 𝑅 ∈ Rels ) )

Proof

Step Hyp Ref Expression
1 dftrrels3 ⊢ TrRels = { 𝑟 ∈ Rels ∣ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) }
2 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑥 𝑟 𝑦 ↔ 𝑥 𝑅 𝑦 ) )
3 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑦 𝑟 𝑧 ↔ 𝑦 𝑅 𝑧 ) )
4 2 3 anbi12d ⊢ ( 𝑟 = 𝑅 → ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) ↔ ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ) )
5 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑥 𝑟 𝑧 ↔ 𝑥 𝑅 𝑧 ) )
6 4 5 imbi12d ⊢ ( 𝑟 = 𝑅 → ( ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ↔ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
7 6 2albidv ⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
8 7 albidv ⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
9 1 8 rabeqel ⊢ ( 𝑅 ∈ TrRels ↔ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ∧ 𝑅 ∈ Rels ) )