Theorem prv

Description: The "proves" relation on a set. A wff encoded as U is true in a model M iff for every valuation s e. ( M ^m _om ) , the interpretation of the wff using the membership relation on M is true. (Contributed by AV, 5-Nov-2023)

	Ref	Expression
Assertion	prv	$$⊢\left(M\in V\wedge U\in W\right)\to \left(M\vDash U\leftrightarrow M{\mathrm{Sat}}_{\in }U=M^{\mathrm{\omega }}\right)$$

Proof

Step	Hyp	Ref	Expression
1		oveq12	$$⊢\left(m=M\wedge u=U\right)\to m{\mathrm{Sat}}_{\in }u=M{\mathrm{Sat}}_{\in }U$$
2		simpl	$$⊢\left(m=M\wedge u=U\right)\to m=M$$
3	2	oveq1d	$$⊢\left(m=M\wedge u=U\right)\to m^{\mathrm{\omega }}=M^{\mathrm{\omega }}$$
4	1 3	eqeq12d	$$⊢\left(m=M\wedge u=U\right)\to \left(m{\mathrm{Sat}}_{\in }u=m^{\mathrm{\omega }}\leftrightarrow M{\mathrm{Sat}}_{\in }U=M^{\mathrm{\omega }}\right)$$
5		df-prv	$$⊢\vDash =\left\{⟨m,u⟩|m{\mathrm{Sat}}_{\in }u=m^{\mathrm{\omega }}\right\}$$
6	4 5	brabga	$$⊢\left(M\in V\wedge U\in W\right)\to \left(M\vDash U\leftrightarrow M{\mathrm{Sat}}_{\in }U=M^{\mathrm{\omega }}\right)$$