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Metamath Proof Explorer

Theorem fmla0

Description: The valid Godel formulas of height 0 is the set of all formulas of the form v_i e. v_j ("Godel-set of membership") coded as <. (/) , <. i , j >. >. . (Contributed by AV, 14-Sep-2023)

Ref Expression
Assertion fmla0 ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) }

Proof

Step Hyp Ref Expression
1 peano1 ⊢ ∅ ∈ ω
2 elelsuc ⊢ ( ∅ ∈ ω → ∅ ∈ suc ω )
3 fmlafv ⊢ ( ∅ ∈ suc ω → ( Fmla ‘ ∅ ) = dom ( ( ∅ Sat ∅ ) ‘ ∅ ) )
4 1 2 3 mp2b ⊢ ( Fmla ‘ ∅ ) = dom ( ( ∅ Sat ∅ ) ‘ ∅ )
5 satf00 ⊢ ( ( ∅ Sat ∅ ) ‘ ∅ ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) }
6 5 dmeqi ⊢ dom ( ( ∅ Sat ∅ ) ‘ ∅ ) = dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) }
7 0ex ⊢ ∅ ∈ V
8 7 isseti ⊢ ∃ 𝑦 𝑦 = ∅
9 19.41v ⊢ ( ∃ 𝑦 ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ( ∃ 𝑦 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) )
10 8 9 mpbiran ⊢ ( ∃ 𝑦 ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) )
11 10 abbii ⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } = { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) }
12 dmopab ⊢ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } = { 𝑥 ∣ ∃ 𝑦 ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) }
13 rabab ⊢ { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) } = { 𝑥 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) }
14 11 12 13 3eqtr4i ⊢ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = ∅ ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ) } = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) }
15 4 6 14 3eqtri ⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) }