Theorem fmla0 35376

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 7868	. . 3 ⊢ ∅ ∈ ω
2		elelsuc 6410	. . 3 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3		fmlafv 35374	. . 3 ⊢ (∅ ∈ suc ω → (Fmla‘∅) = dom ((∅ Sat ∅)‘∅))
4	1, 2, 3	mp2b 10	. 2 ⊢ (Fmla‘∅) = dom ((∅ Sat ∅)‘∅)
5		satf00 35368	. . 3 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
6	5	dmeqi 5871	. 2 ⊢ dom ((∅ Sat ∅)‘∅) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
7		0ex 5265	. . . . . 6 ⊢ ∅ ∈ V
8	7	isseti 3468	. . . . 5 ⊢ ∃𝑦 𝑦 = ∅
9		19.41v 1949	. . . . 5 ⊢ (∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ (∃𝑦 𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)))
10	8, 9	mpbiran 709	. . . 4 ⊢ (∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))
11	10	abbii 2797	. . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
12		dmopab 5882	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
13		rabab 3481	. . 3 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
14	11, 12, 13	3eqtr4i 2763	. 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 2757	1 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∃wrex 3054  {crab 3408  Vcvv 3450  ∅c0 4299  {copab 5172  dom cdm 5641  suc csuc 6337  ‘cfv 6514  (class class class)co 7390  ωcom 7845  ∈_𝑔cgoe 35327   Sat csat 35330  Fmlacfmla 35331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-map 8804  df-goel 35334  df-sat 35337  df-fmla 35339

This theorem is referenced by:  fmla0xp  35377  fmlafvel  35379  fmla1  35381  fmlaomn0  35384  gonan0  35386  goaln0  35387  gonar  35389  goalr  35391  fmla0disjsuc  35392  satfv0fvfmla0  35407  sategoelfvb  35413