Theorem fmlasuc 35859

Description: The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)

Assertion

Ref	Expression
fmlasuc	⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))

Distinct variable group:   𝑢,𝑁,𝑣,𝑥,𝑖

Proof of Theorem fmlasuc

Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmlasuc0 35857	. 2 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))}))
2		eqid 2763	. . . . . . . 8 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
3	2	satf0op 35850	. . . . . . 7 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
5	4	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
6	5	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤))))
7	6	cbvrexvw 3244	. . . . . . . . 9 ⊢ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
8	7	orbi1i 926	. . . . . . . 8 ⊢ ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
9		fmlafvel 35858	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑧 ∈ (Fmla‘𝑁) ↔ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
10	9	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑧 ∈ (Fmla‘𝑁)))
11	10	adantld 495	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑧 ∈ (Fmla‘𝑁)))
12	11	imp 411	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑧 ∈ (Fmla‘𝑁))
13		vex 3459	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
14		0ex 5271	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15	13, 14	op1std 7997	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (1st ‘𝑦) = 𝑧)
16	15	eleq1d 2848	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
17	16	ad2antrl 740	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
18	12, 17	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑦) ∈ (Fmla‘𝑁))
19	18	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))) → (1st ‘𝑦) ∈ (Fmla‘𝑁))
20		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → (𝑢⊼𝑔𝑣) = ((1st ‘𝑦)⊼𝑔𝑣))
21	20	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = (𝑢⊼𝑔𝑣) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
22	21	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
23		eqidd 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑖 = 𝑖)
24		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑢 = (1st ‘𝑦))
25	23, 24	goaleq12d 35824	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → ∀𝑔𝑖𝑢 = ∀𝑔𝑖(1st ‘𝑦))
26	25	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = ∀𝑔𝑖𝑢 ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
27	26	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
28	22, 27	orbi12d 931	. . . . . . . . . . . 12 ⊢ (𝑢 = (1st ‘𝑦) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))))
29	28	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))) ∧ 𝑢 = (1st ‘𝑦)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))))
30	2	satf0op 35850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31		fmlafvel 35858	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (Fmla‘𝑁) ↔ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
32	31	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ω → (⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑦 ∈ (Fmla‘𝑁)))
33	32	adantld 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑦 ∈ (Fmla‘𝑁)))
34	33	imp 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑦 ∈ (Fmla‘𝑁))
35		vex 3459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
36	35, 14	op1std 7997	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → (1st ‘𝑤) = 𝑦)
37	36	eleq1d 2848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
38	37	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
39	34, 38	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
40	39	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
41		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (1st ‘𝑤) → (𝑧⊼𝑔𝑣) = (𝑧⊼𝑔(1st ‘𝑤)))
42	41	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (1st ‘𝑤) → (𝑥 = (𝑧⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
43	42	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) ∧ 𝑣 = (1st ‘𝑤)) → (𝑥 = (𝑧⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
44		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → 𝑥 = (𝑧⊼𝑔(1st ‘𝑤)))
45	40, 43, 44	rspcedvd 3584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))
46	45	exp31 424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
47	46	exlimdv 1963	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
48	30, 47	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
49	48	rexlimdv 3164	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
50	49	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
51	15	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) = (𝑧⊼𝑔(1st ‘𝑤)))
52	51	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
53	52	rexbidv 3189	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
54	15	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔𝑣) = (𝑧⊼𝑔𝑣))
55	54	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔𝑣)))
56	55	rexbidv 3189	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
57	53, 56	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
58	57	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)) ↔ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
59	50, 58	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
60	59	orim1d 981	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))))
61	60	3impia 1135	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
62	19, 29, 61	rspcedvd 3584	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) ∧ (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
63	62	3exp 1137	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
64	63	exlimdv 1963	. . . . . . . 8 ⊢ (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
65	8, 64	syl7bi 258	. . . . . . 7 ⊢ (𝑁 ∈ ω → (∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
66	3, 65	sylbid 243	. . . . . 6 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))))
67	66	rexlimdv 3164	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) → ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
68		fmlafvel 35858	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑢 ∈ (Fmla‘𝑁) ↔ ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
69	68	biimpa 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
70	69	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
71		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
72	71, 14	op1std 7997	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (1st ‘𝑦) = 𝑢)
73	72	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔(1st ‘𝑧)))
74	73	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
75	74	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
76		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
77	76, 72	goaleq12d 35824	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ∀𝑔𝑖(1st ‘𝑦) = ∀𝑔𝑖𝑢)
78	77	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ 𝑥 = ∀𝑔𝑖𝑢))
79	78	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
80	75, 79	orbi12d 931	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
81	80	adantl 486	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) ∧ 𝑦 = ⟨𝑢, ∅⟩) → ((∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) ↔ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
82		fmlafvel 35858	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) ↔ ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
83	82	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
84	83	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
85	84	imp 411	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
86	85	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
87		vex 3459	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
88	87, 14	op1std 7997	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (1st ‘𝑧) = 𝑣)
89	88	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑢⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔𝑣))
90	89	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
91	90	adantl 486	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) ∧ 𝑧 = ⟨𝑣, ∅⟩) → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
92		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → 𝑥 = (𝑢⊼𝑔𝑣))
93	86, 91, 92	rspcedvd 3584	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)))
94	93	rexlimdva2 3168	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
95	94	orim1d 981	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ((∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
96	95	imp 411	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
97	70, 81, 96	rspcedvd 3584	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
98	97	rexlimdva2 3168	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) → ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))))
99	67, 98	impbid 215	. . . 4 ⊢ (𝑁 ∈ ω → (∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)) ↔ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
100	99	abbidv 2829	. . 3 ⊢ (𝑁 ∈ ω → {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))} = {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})
101	100	uneq2d 4123	. 2 ⊢ (𝑁 ∈ ω → ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑦 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦))}) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
102	1, 101	eqtrd 2798	1 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∃wrex 3089   ∪ cun 3904  ∅c0 4287  ⟨cop 4596  suc csuc 6364  ‘cfv 6538  (class class class)co 7412  ωcom 7863  1^st c1st 7985  ⊼_𝑔cgna 35807  ∀_𝑔cgol 35808   Sat csat 35809  Fmlacfmla 35810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-map 8827  df-goel 35813  df-goal 35815  df-sat 35816  df-fmla 35818

This theorem is referenced by:  fmla1  35860  isfmlasuc  35861  fmlasssuc  35862  fmlaomn0  35863