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Metamath Proof Explorer

Theorem eleqvrels3

Description: Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021)

Ref Expression
Assertion eleqvrels3 ( 𝑅 ∈ EqvRels ↔ ( ( ∀ 𝑥 ∈ dom 𝑅 𝑥 𝑅 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) ∧ 𝑅 ∈ Rels ) )

Proof

Step Hyp Ref Expression
1 dfeqvrels3 ⊢ EqvRels = { 𝑟 ∈ Rels ∣ ( ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ) }
2 dmeq ⊢ ( 𝑟 = 𝑅 → dom 𝑟 = dom 𝑅 )
3 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑥 𝑟 𝑥 ↔ 𝑥 𝑅 𝑥 ) )
4 2 3 raleqbidv ⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ↔ ∀ 𝑥 ∈ dom 𝑅 𝑥 𝑅 𝑥 ) )
5 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑥 𝑟 𝑦 ↔ 𝑥 𝑅 𝑦 ) )
6 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑦 𝑟 𝑥 ↔ 𝑦 𝑅 𝑥 ) )
7 5 6 imbi12d ⊢ ( 𝑟 = 𝑅 → ( ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ↔ ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ) )
8 7 2albidv ⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ) )
9 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑦 𝑟 𝑧 ↔ 𝑦 𝑅 𝑧 ) )
10 5 9 anbi12d ⊢ ( 𝑟 = 𝑅 → ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) ↔ ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ) )
11 breq ⊢ ( 𝑟 = 𝑅 → ( 𝑥 𝑟 𝑧 ↔ 𝑥 𝑅 𝑧 ) )
12 10 11 imbi12d ⊢ ( 𝑟 = 𝑅 → ( ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ↔ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
13 12 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 ) ) )
14 13 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 ) ) )
15 4 8 14 3anbi123d ⊢ ( 𝑟 = 𝑅 → ( ( ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ) ↔ ( ∀ 𝑥 ∈ dom 𝑅 𝑥 𝑅 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) ) )
16 1 15 rabeqel ⊢ ( 𝑅 ∈ EqvRels ↔ ( ( ∀ 𝑥 ∈ dom 𝑅 𝑥 𝑅 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) ∧ 𝑅 ∈ Rels ) )