Metamath Proof Explorer


Theorem satfvel

Description: An element of the value of the satisfaction predicate as function over wff codes in the model M and the binary relation E on M at the code U for a wff using e. , -/\ , A. is a valuation S :om --> M of the variables (v0 = ( S(/) ) , v_1 = ( S1o ) , etc.) so that U is true under the assignment S . (Contributed by AV, 29-Oct-2023)

Ref Expression
Assertion satfvel M V E W U Fmla ω S M Sat E ω U S : ω M

Proof

Step Hyp Ref Expression
1 satfun M V E W M Sat E ω : Fmla ω 𝒫 M ω
2 ffvelrn M Sat E ω : Fmla ω 𝒫 M ω U Fmla ω M Sat E ω U 𝒫 M ω
3 fvex M Sat E ω U V
4 3 elpw M Sat E ω U 𝒫 M ω M Sat E ω U M ω
5 ssel M Sat E ω U M ω S M Sat E ω U S M ω
6 elmapi S M ω S : ω M
7 5 6 syl6 M Sat E ω U M ω S M Sat E ω U S : ω M
8 4 7 sylbi M Sat E ω U 𝒫 M ω S M Sat E ω U S : ω M
9 2 8 syl M Sat E ω : Fmla ω 𝒫 M ω U Fmla ω S M Sat E ω U S : ω M
10 9 ex M Sat E ω : Fmla ω 𝒫 M ω U Fmla ω S M Sat E ω U S : ω M
11 1 10 syl M V E W U Fmla ω S M Sat E ω U S : ω M
12 11 3imp M V E W U Fmla ω S M Sat E ω U S : ω M