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Metamath Proof Explorer

Theorem trrelressn

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 ) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024)

Ref Expression
Assertion trrelressn ⊢ TrRel R ↾ A

Proof

Step Hyp Ref Expression
1 trressn ⊢ ∀ x ∀ y ∀ z x R ↾ A y ∧ y R ↾ A z → x R ↾ A z
2 relres ⊢ Rel ⁡ R ↾ A
3 dftrrel3 ⊢ TrRel R ↾ A ↔ ∀ x ∀ y ∀ z x R ↾ A y ∧ y R ↾ A z → x R ↾ A z ∧ Rel ⁡ R ↾ A
4 1 2 3 mpbir2an ⊢ TrRel R ↾ A