Theorem fmla0 35025

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 7900	. . 3 ⊢ ∅ ∈ ω
2		elelsuc 6447	. . 3 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3		fmlafv 35023	. . 3 ⊢ (∅ ∈ suc ω → (Fmla‘∅) = dom ((∅ Sat ∅)‘∅))
4	1, 2, 3	mp2b 10	. 2 ⊢ (Fmla‘∅) = dom ((∅ Sat ∅)‘∅)
5		satf00 35017	. . 3 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
6	5	dmeqi 5911	. 2 ⊢ dom ((∅ Sat ∅)‘∅) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
7		0ex 5311	. . . . . 6 ⊢ ∅ ∈ V
8	7	isseti 3489	. . . . 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 707	. . . 4 ⊢ (∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))
11	10	abbii 2798	. . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
12		dmopab 5922	. . 3 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} = {𝑥 ∣ ∃𝑦(𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
13		rabab 3502	. . 3 ⊢ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
14	11, 12, 13	3eqtr4i 2766	. 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 2760	1 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}

Syntax hints:   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2705  ∃wrex 3067  {crab 3430  Vcvv 3473  ∅c0 4326  {copab 5214  dom cdm 5682  suc csuc 6376  ‘cfv 6553  (class class class)co 7426  ωcom 7876  ∈_𝑔cgoe 34976   Sat csat 34979  Fmlacfmla 34980

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-map 8853  df-goel 34983  df-sat 34986  df-fmla 34988

This theorem is referenced by:  fmla0xp  35026  fmlafvel  35028  fmla1  35030  fmlaomn0  35033  gonan0  35035  goaln0  35036  gonar  35038  goalr  35040  fmla0disjsuc  35041  satfv0fvfmla0  35056  sategoelfvb  35062

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