Metamath Proof Explorer


Theorem satef

Description: The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023)

Ref Expression
Assertion satef ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( 𝑀 Sat 𝑈 ) ) → 𝑆 : ω ⟶ 𝑀 )

Proof

Step Hyp Ref Expression
1 satefv ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀 Sat 𝑈 ) = ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) )
2 1 eleq2d ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( 𝑀 Sat 𝑈 ) ↔ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) )
3 simpl ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → 𝑀𝑉 )
4 incom ( E ∩ ( 𝑀 × 𝑀 ) ) = ( ( 𝑀 × 𝑀 ) ∩ E )
5 sqxpexg ( 𝑀𝑉 → ( 𝑀 × 𝑀 ) ∈ V )
6 5 adantr ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀 × 𝑀 ) ∈ V )
7 inex1g ( ( 𝑀 × 𝑀 ) ∈ V → ( ( 𝑀 × 𝑀 ) ∩ E ) ∈ V )
8 6 7 syl ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( ( 𝑀 × 𝑀 ) ∩ E ) ∈ V )
9 4 8 eqeltrid ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V )
10 3 9 jca ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) )
11 10 adantr ( ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) )
12 simpr ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → 𝑈 ∈ ( Fmla ‘ ω ) )
13 12 adantr ( ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑈 ∈ ( Fmla ‘ ω ) )
14 simpr ( ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) )
15 11 13 14 3jca ( ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → ( ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) )
16 15 ex ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) → ( ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) )
17 2 16 sylbid ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( 𝑀 Sat 𝑈 ) → ( ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) )
18 17 3impia ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( 𝑀 Sat 𝑈 ) ) → ( ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) )
19 satfvel ( ( ( 𝑀𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑆 : ω ⟶ 𝑀 )
20 18 19 syl ( ( 𝑀𝑉𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( 𝑀 Sat 𝑈 ) ) → 𝑆 : ω ⟶ 𝑀 )