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	$$⊢TrRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		trressn	$$⊢\forall x\forall y\forall z\left(\left(x\left({R\upharpoonright }_{\left\{A\right\}}\right)y\wedge y\left({R\upharpoonright }_{\left\{A\right\}}\right)z\right)\to x\left({R\upharpoonright }_{\left\{A\right\}}\right)z\right)$$
2		relres	$$⊢\mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)$$
3		dftrrel3	$$⊢TrRel\left({R\upharpoonright }_{\left\{A\right\}}\right)\leftrightarrow \left(\forall x\forall y\forall z\left(\left(x\left({R\upharpoonright }_{\left\{A\right\}}\right)y\wedge y\left({R\upharpoonright }_{\left\{A\right\}}\right)z\right)\to x\left({R\upharpoonright }_{\left\{A\right\}}\right)z\right)\wedge \mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)\right)$$
4	1 2 3	mpbir2an	$$⊢TrRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$