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 | ⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊 ) → ( 𝑀 ⊧ 𝑈 ↔ ( 𝑀 Sat∈ 𝑈 ) = ( 𝑀 ↑m ω ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 | ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑢 = 𝑈 ) → ( 𝑚 Sat∈ 𝑢 ) = ( 𝑀 Sat∈ 𝑈 ) ) | |
2 | simpl | ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑢 = 𝑈 ) → 𝑚 = 𝑀 ) | |
3 | 2 | oveq1d | ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑢 = 𝑈 ) → ( 𝑚 ↑m ω ) = ( 𝑀 ↑m ω ) ) |
4 | 1 3 | eqeq12d | ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑢 = 𝑈 ) → ( ( 𝑚 Sat∈ 𝑢 ) = ( 𝑚 ↑m ω ) ↔ ( 𝑀 Sat∈ 𝑈 ) = ( 𝑀 ↑m ω ) ) ) |
5 | df-prv | ⊢ ⊧ = { 〈 𝑚 , 𝑢 〉 ∣ ( 𝑚 Sat∈ 𝑢 ) = ( 𝑚 ↑m ω ) } | |
6 | 4 5 | brabga | ⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊 ) → ( 𝑀 ⊧ 𝑈 ↔ ( 𝑀 Sat∈ 𝑈 ) = ( 𝑀 ↑m ω ) ) ) |