Theorem fmla 35563

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 35527	. . 3 ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
2	1	fveq1i 6841	. 2 ⊢ (Fmla‘ω) = ((𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))‘ω)
3		omex 9564	. . 3 ⊢ ω ∈ V
4		eqidd 2737	. . . 4 ⊢ (ω ∈ V → (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)))
5		fveq2 6840	. . . . . 6 ⊢ (𝑛 = ω → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘ω))
6	5	dmeqd 5860	. . . . 5 ⊢ (𝑛 = ω → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘ω))
7	6	adantl 481	. . . 4 ⊢ ((ω ∈ V ∧ 𝑛 = ω) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘ω))
8		sucidg 6406	. . . 4 ⊢ (ω ∈ V → ω ∈ suc ω)
9		fvex 6853	. . . . . 6 ⊢ ((∅ Sat ∅)‘ω) ∈ V
10	9	dmex 7860	. . . . 5 ⊢ dom ((∅ Sat ∅)‘ω) ∈ V
11	10	a1i 11	. . . 4 ⊢ (ω ∈ V → dom ((∅ Sat ∅)‘ω) ∈ V)
12	4, 7, 8, 11	fvmptd 6955	. . 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 6407	. . . . . 6 ⊢ ω ∈ suc ω
15		satf0sucom 35555	. . . . . 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 7833	. . . . . 6 ⊢ Lim ω
18		rdglim2a 8372	. . . . . 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 693	. . . . 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 2759	. . . 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 5859	. . 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 5868	. . 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 6398	. . . . . 6 ⊢ (𝑛 ∈ ω → 𝑛 ∈ suc ω)
24		fmlafv 35562	. . . . . 6 ⊢ (𝑛 ∈ suc ω → (Fmla‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
25	23, 24	syl 17	. . . . 5 ⊢ (𝑛 ∈ ω → (Fmla‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
26		satf0sucom 35555	. . . . . . 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 17	. . . . . 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 5860	. . . . 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 2772	. . . 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 4955	. . 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 2763	. 2 ⊢ dom ((∅ Sat ∅)‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
32	2, 13, 31	3eqtri 2763	1 ⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)

Syntax hints:   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429   ∪ cun 3887  ∅c0 4273  ∪ ciun 4933  {copab 5147   ↦ cmpt 5166  dom cdm 5631  Lim wlim 6324  suc csuc 6325  ‘cfv 6498  (class class class)co 7367  ωcom 7817  1^st c1st 7940  reccrdg 8348  ∈_𝑔cgoe 35515  ⊼_𝑔cgna 35516  ∀_𝑔cgol 35517   Sat csat 35518  Fmlacfmla 35519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-map 8775  df-sat 35525  df-fmla 35527

This theorem is referenced by:  fmlan0  35573  satfun  35593