Metamath Proof Explorer


Theorem sategoelfvb

Description: Characterization of a valuation S of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023)

Ref Expression
Hypothesis sategoelfvb.s 𝐸 = ( 𝑀 Sat ( 𝐴𝑔 𝐵 ) )
Assertion sategoelfvb ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆𝐸 ↔ ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆𝐴 ) ∈ ( 𝑆𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 sategoelfvb.s 𝐸 = ( 𝑀 Sat ( 𝐴𝑔 𝐵 ) )
2 ovexd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴𝑔 𝐵 ) ∈ V )
3 simpl ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω )
4 opeq1 ( 𝑎 = 𝐴 → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝑏 ⟩ )
5 4 opeq2d ( 𝑎 = 𝐴 → ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ )
6 5 eqeq2d ( 𝑎 = 𝐴 → ( ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ ↔ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ ) )
7 6 rexbidv ( 𝑎 = 𝐴 → ( ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ ↔ ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ ) )
8 7 adantl ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝑎 = 𝐴 ) → ( ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ ↔ ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ ) )
9 simpr ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐵 ∈ ω )
10 opeq2 ( 𝑏 = 𝐵 → ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
11 10 opeq2d ( 𝑏 = 𝐵 → ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ )
12 11 eqeq2d ( 𝑏 = 𝐵 → ( ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ ↔ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) )
13 12 adantl ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝑏 = 𝐵 ) → ( ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ ↔ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) )
14 eqidd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ )
15 9 13 14 rspcedvd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝐴 , 𝑏 ⟩ ⟩ )
16 3 8 15 rspcedvd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ )
17 goel ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴𝑔 𝐵 ) = ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ )
18 goel ( ( 𝑎 ∈ ω ∧ 𝑏 ∈ ω ) → ( 𝑎𝑔 𝑏 ) = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ )
19 17 18 eqeqan12d ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ ( 𝑎 ∈ ω ∧ 𝑏 ∈ ω ) ) → ( ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) ↔ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ ) )
20 19 2rexbidva ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) ↔ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ = ⟨ ∅ , ⟨ 𝑎 , 𝑏 ⟩ ⟩ ) )
21 16 20 mpbird ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) )
22 eqeq1 ( 𝑥 = ( 𝐴𝑔 𝐵 ) → ( 𝑥 = ( 𝑎𝑔 𝑏 ) ↔ ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) ) )
23 22 2rexbidv ( 𝑥 = ( 𝐴𝑔 𝐵 ) → ( ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 𝑥 = ( 𝑎𝑔 𝑏 ) ↔ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) ) )
24 fmla0 ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 𝑥 = ( 𝑎𝑔 𝑏 ) }
25 23 24 elrab2 ( ( 𝐴𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) ↔ ( ( 𝐴𝑔 𝐵 ) ∈ V ∧ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴𝑔 𝐵 ) = ( 𝑎𝑔 𝑏 ) ) )
26 2 21 25 sylanbrc ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) )
27 satefvfmla0 ( ( 𝑀𝑉 ∧ ( 𝐴𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) ) → ( 𝑀 Sat ( 𝐴𝑔 𝐵 ) ) = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) } )
28 26 27 sylan2 ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑀 Sat ( 𝐴𝑔 𝐵 ) ) = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) } )
29 1 28 syl5eq ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → 𝐸 = { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) } )
30 29 eleq2d ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆𝐸𝑆 ∈ { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) } ) )
31 fveq1 ( 𝑎 = 𝑆 → ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) )
32 fveq1 ( 𝑎 = 𝑆 → ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) )
33 31 32 eleq12d ( 𝑎 = 𝑆 → ( ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ↔ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ) )
34 33 elrab ( 𝑆 ∈ { 𝑎 ∈ ( 𝑀m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) } ↔ ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ) )
35 30 34 bitrdi ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆𝐸 ↔ ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ) ) )
36 17 fveq2d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) = ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) )
37 36 fveq2d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) = ( 1st ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) )
38 0ex ∅ ∈ V
39 opex 𝐴 , 𝐵 ⟩ ∈ V
40 38 39 op2nd ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) = ⟨ 𝐴 , 𝐵
41 40 fveq2i ( 1st ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) = ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ )
42 op1stg ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴 )
43 41 42 syl5eq ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) = 𝐴 )
44 37 43 eqtrd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) = 𝐴 )
45 44 fveq2d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) = ( 𝑆𝐴 ) )
46 36 fveq2d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) = ( 2nd ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) )
47 40 fveq2i ( 2nd ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) = ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ )
48 op2ndg ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐵 )
49 47 48 syl5eq ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ⟨ ∅ , ⟨ 𝐴 , 𝐵 ⟩ ⟩ ) ) = 𝐵 )
50 46 49 eqtrd ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) = 𝐵 )
51 50 fveq2d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) = ( 𝑆𝐵 ) )
52 45 51 eleq12d ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ↔ ( 𝑆𝐴 ) ∈ ( 𝑆𝐵 ) ) )
53 52 adantl ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ↔ ( 𝑆𝐴 ) ∈ ( 𝑆𝐵 ) ) )
54 53 anbi2d ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴𝑔 𝐵 ) ) ) ) ) ↔ ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆𝐴 ) ∈ ( 𝑆𝐵 ) ) ) )
55 35 54 bitrd ( ( 𝑀𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆𝐸 ↔ ( 𝑆 ∈ ( 𝑀m ω ) ∧ ( 𝑆𝐴 ) ∈ ( 𝑆𝐵 ) ) ) )