Theorem fmla0 34891

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

Assertion

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

Distinct variable group:   𝑖,𝑗,𝑥

Proof of Theorem fmla0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7873	. . 3 ⊢ ∅ ∈ ω
2		elelsuc 6428	. . 3 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3		fmlafv 34889	. . 3 ⊢ (∅ ∈ suc ω → (Fmla‘∅) = dom ((∅ Sat ∅)‘∅))
4	1, 2, 3	mp2b 10	. 2 ⊢ (Fmla‘∅) = dom ((∅ Sat ∅)‘∅)
5		satf00 34883	. . 3 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
6	5	dmeqi 5895	. 2 ⊢ dom ((∅ Sat ∅)‘∅) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
7		0ex 5298	. . . . . 6 ⊢ ∅ ∈ V
8	7	isseti 3482	. . . . 5 ⊢ ∃𝑦 𝑦 = ∅
9		19.41v 1945	. . . . 5 ⊢ (∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ (∃𝑦 𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)))
10	8, 9	mpbiran 706	. . . 4 ⊢ (∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))
11	10	abbii 2794	. . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
12		dmopab 5906	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
13		rabab 3495	. . 3 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
14	11, 12, 13	3eqtr4i 2762	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
15	4, 6, 14	3eqtri 2756	1 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}

Syntax hints:   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701  ∃wrex 3062  {crab 3424  Vcvv 3466  ∅c0 4315  {copab 5201  dom cdm 5667  suc csuc 6357  ‘cfv 6534  (class class class)co 7402  ωcom 7849  ∈_𝑔cgoe 34842   Sat csat 34845  Fmlacfmla 34846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-map 8819  df-goel 34849  df-sat 34852  df-fmla 34854

This theorem is referenced by:  fmla0xp  34892  fmlafvel  34894  fmla1  34896  fmlaomn0  34899  gonan0  34901  goaln0  34902  gonar  34904  goalr  34906  fmla0disjsuc  34907  satfv0fvfmla0  34922  sategoelfvb  34928