Theorem fmla 36115

Description: The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)

Assertion

Ref	Expression
fmla	⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)

Proof of Theorem fmla

Dummy variables 𝑓 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fmla 36079	. . 3 ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
2	1	fveq1i 6878	. 2 ⊢ (Fmla‘ω) = ((𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))‘ω)
3		omex 9628	. . 3 ⊢ ω ∈ V
4		eqidd 2762	. . . 4 ⊢ (ω ∈ V → (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)))
5		fveq2 6877	. . . . . 6 ⊢ (𝑛 = ω → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘ω))
6	5	dmeqd 5887	. . . . 5 ⊢ (𝑛 = ω → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘ω))
7	6	adantl 487	. . . 4 ⊢ ((ω ∈ V ∧ 𝑛 = ω) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘ω))
8		sucidg 6439	. . . 4 ⊢ (ω ∈ V → ω ∈ suc ω)
9		fvex 6890	. . . . . 6 ⊢ ((∅ Sat ∅)‘ω) ∈ V
10	9	dmex 7910	. . . . 5 ⊢ dom ((∅ Sat ∅)‘ω) ∈ V
11	10	a1i 11	. . . 4 ⊢ (ω ∈ V → dom ((∅ Sat ∅)‘ω) ∈ V)
12	4, 7, 8, 11	fvmptd 6993	. . 3 ⊢ (ω ∈ V → ((𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))‘ω) = dom ((∅ Sat ∅)‘ω))
13	3, 12	ax-mp 5	. 2 ⊢ ((𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))‘ω) = dom ((∅ Sat ∅)‘ω)
14	3	sucid 6440	. . . . . 6 ⊢ ω ∈ suc ω
15		satf0sucom 36107	. . . . . 6 ⊢ (ω ∈ suc ω → ((∅ Sat ∅)‘ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘ω))
16	14, 15	ax-mp 5	. . . . 5 ⊢ ((∅ Sat ∅)‘ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘ω)
17		limom 7882	. . . . . 6 ⊢ Lim ω
18		rdglim2a 8425	. . . . . 6 ⊢ ((ω ∈ V ∧ Lim ω) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘ω) = ∪ 𝑛 ∈ ω (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛))
19	3, 17, 18	mp2an 705	. . . . 5 ⊢ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘ω) = ∪ 𝑛 ∈ ω (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛)
20	16, 19	eqtri 2784	. . . 4 ⊢ ((∅ Sat ∅)‘ω) = ∪ 𝑛 ∈ ω (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛)
21	20	dmeqi 5886	. . 3 ⊢ dom ((∅ Sat ∅)‘ω) = dom ∪ 𝑛 ∈ ω (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛)
22		dmiun 5895	. . 3 ⊢ dom ∪ 𝑛 ∈ ω (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛) = ∪ 𝑛 ∈ ω dom (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛)
23		elelsuc 6431	. . . . . 6 ⊢ (𝑛 ∈ ω → 𝑛 ∈ suc ω)
24		fmlafv 36114	. . . . . 6 ⊢ (𝑛 ∈ suc ω → (Fmla‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
25	23, 24	syl 18	. . . . 5 ⊢ (𝑛 ∈ ω → (Fmla‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
26		satf0sucom 36107	. . . . . . 7 ⊢ (𝑛 ∈ suc ω → ((∅ Sat ∅)‘𝑛) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛))
27	23, 26	syl 18	. . . . . 6 ⊢ (𝑛 ∈ ω → ((∅ Sat ∅)‘𝑛) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛))
28	27	dmeqd 5887	. . . . 5 ⊢ (𝑛 ∈ ω → dom ((∅ Sat ∅)‘𝑛) = dom (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛))
29	25, 28	eqtr2d 2797	. . . 4 ⊢ (𝑛 ∈ ω → dom (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛) = (Fmla‘𝑛))
30	29	iuneq2i 4973	. . 3 ⊢ ∪ 𝑛 ∈ ω dom (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑛) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
31	21, 22, 30	3eqtri 2788	. 2 ⊢ dom ((∅ Sat ∅)‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
32	2, 13, 31	3eqtri 2788	1 ⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451   ∪ cun 3897  ∅c0 4279  ∪ ciun 4951  {copab 5167   ↦ cmpt 5186  dom cdm 5651  Lim wlim 6356  suc csuc 6357  ‘cfv 6531  (class class class)co 7412  ωcom 7866  1^st c1st 7988  reccrdg 8401  ∈_𝑔cgoe 36067  ⊼_𝑔cgna 36068  ∀_𝑔cgol 36069   Sat csat 36070  Fmlacfmla 36071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-map 8833  df-sat 36077  df-fmla 36079

This theorem is used by:  fmlan0  36125  satfun  36145