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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 } )