Metamath Proof Explorer


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 ( ( 𝑀𝑉𝑈𝑊 ) → ( 𝑀𝑈 ↔ ( 𝑀 Sat 𝑈 ) = ( 𝑀m ω ) ) )

Proof

Step Hyp Ref Expression
1 oveq12 ( ( 𝑚 = 𝑀𝑢 = 𝑈 ) → ( 𝑚 Sat 𝑢 ) = ( 𝑀 Sat 𝑈 ) )
2 simpl ( ( 𝑚 = 𝑀𝑢 = 𝑈 ) → 𝑚 = 𝑀 )
3 2 oveq1d ( ( 𝑚 = 𝑀𝑢 = 𝑈 ) → ( 𝑚m ω ) = ( 𝑀m ω ) )
4 1 3 eqeq12d ( ( 𝑚 = 𝑀𝑢 = 𝑈 ) → ( ( 𝑚 Sat 𝑢 ) = ( 𝑚m ω ) ↔ ( 𝑀 Sat 𝑈 ) = ( 𝑀m ω ) ) )
5 df-prv ⊧ = { ⟨ 𝑚 , 𝑢 ⟩ ∣ ( 𝑚 Sat 𝑢 ) = ( 𝑚m ω ) }
6 4 5 brabga ( ( 𝑀𝑉𝑈𝑊 ) → ( 𝑀𝑈 ↔ ( 𝑀 Sat 𝑈 ) = ( 𝑀m ω ) ) )