Theorem fmlasuc 33714

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 33712	. 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 2737	. . . . . . . 8 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
3	2	satf0op 33705	. . . . . . 7 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4		fveq2 6837	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
5	4	oveq2d 7365	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
6	5	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤))))
7	6	cbvrexvw 3224	. . . . . . . . 9 ⊢ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
8	7	orbi1i 912	. . . . . . . 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 33713	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑧 ∈ (Fmla‘𝑁) ↔ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
10	9	biimprd 248	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑧 ∈ (Fmla‘𝑁)))
11	10	adantld 492	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑧 ∈ (Fmla‘𝑁)))
12	11	imp 408	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑧 ∈ (Fmla‘𝑁))
13		vex 3447	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
14		0ex 5262	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15	13, 14	op1std 7921	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (1st ‘𝑦) = 𝑧)
16	15	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
17	16	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
18	12, 17	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑦) ∈ (Fmla‘𝑁))
19	18	3adant3 1132	. . . . . . . . . . 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 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → (𝑢⊼𝑔𝑣) = ((1st ‘𝑦)⊼𝑔𝑣))
21	20	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = (𝑢⊼𝑔𝑣) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
22	21	rexbidv 3173	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
23		eqidd 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑖 = 𝑖)
24		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑢 = (1st ‘𝑦))
25	23, 24	goaleq12d 33679	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → ∀𝑔𝑖𝑢 = ∀𝑔𝑖(1st ‘𝑦))
26	25	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = ∀𝑔𝑖𝑢 ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
27	26	rexbidv 3173	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
28	22, 27	orbi12d 917	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 33705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31		fmlafvel 33713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (Fmla‘𝑁) ↔ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
32	31	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ω → (⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑦 ∈ (Fmla‘𝑁)))
33	32	adantld 492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑦 ∈ (Fmla‘𝑁)))
34	33	imp 408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑦 ∈ (Fmla‘𝑁))
35		vex 3447	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
36	35, 14	op1std 7921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → (1st ‘𝑤) = 𝑦)
37	36	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
38	37	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
39	34, 38	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
40	39	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
41		oveq2 7357	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (1st ‘𝑤) → (𝑧⊼𝑔𝑣) = (𝑧⊼𝑔(1st ‘𝑤)))
42	41	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (1st ‘𝑤) → (𝑥 = (𝑧⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
43	42	adantl 483	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → 𝑥 = (𝑧⊼𝑔(1st ‘𝑤)))
45	40, 43, 44	rspcedvd 3581	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))
46	45	exp31 421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
47	46	exlimdv 1936	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
48	30, 47	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
49	48	rexlimdv 3148	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
50	49	adantr 482	. . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) = (𝑧⊼𝑔(1st ‘𝑤)))
52	51	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
53	52	rexbidv 3173	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
54	15	oveq1d 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔𝑣) = (𝑧⊼𝑔𝑣))
55	54	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔𝑣)))
56	55	rexbidv 3173	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
57	53, 56	imbi12d 345	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 964	. . . . . . . . . . . 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 1117	. . . . . . . . . . 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 3581	. . . . . . . . . 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 1119	. . . . . . . . 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 1936	. . . . . . . 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 255	. . . . . . 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 239	. . . . . 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 3148	. . . . 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 33713	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑢 ∈ (Fmla‘𝑁) ↔ ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
69	68	biimpa 478	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
70	69	adantr 482	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
71		vex 3447	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
72	71, 14	op1std 7921	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (1st ‘𝑦) = 𝑢)
73	72	oveq1d 7364	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔(1st ‘𝑧)))
74	73	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
75	74	rexbidv 3173	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
76		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
77	76, 72	goaleq12d 33679	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ∀𝑔𝑖(1st ‘𝑦) = ∀𝑔𝑖𝑢)
78	77	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ 𝑥 = ∀𝑔𝑖𝑢))
79	78	rexbidv 3173	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
80	75, 79	orbi12d 917	. . . . . . . 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 483	. . . . . . 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 33713	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) ↔ ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
83	82	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
84	83	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
85	84	imp 408	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
86	85	adantr 482	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
87		vex 3447	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
88	87, 14	op1std 7921	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (1st ‘𝑧) = 𝑣)
89	88	oveq2d 7365	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑢⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔𝑣))
90	89	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
91	90	adantl 483	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) ∧ 𝑧 = ⟨𝑣, ∅⟩) → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
92		simpr 486	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → 𝑥 = (𝑢⊼𝑔𝑣))
93	86, 91, 92	rspcedvd 3581	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)))
94	93	rexlimdva2 3152	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
95	94	orim1d 964	. . . . . . . 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 408	. . . . . . 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 3581	. . . . . 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 3152	. . . . 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 211	. . . 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 2806	. . 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 4121	. 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 2777	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 205   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2714  ∃wrex 3071   ∪ cun 3906  ∅c0 4280  ⟨cop 4590  suc csuc 6315  ‘cfv 6491  (class class class)co 7349  ωcom 7792  1^st c1st 7909  ⊼_𝑔cgna 33662  ∀_𝑔cgol 33663   Sat csat 33664  Fmlacfmla 33665

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7662  ax-inf2 9510

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5528  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5585  df-we 5587  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-pred 6249  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7793  df-1st 7911  df-2nd 7912  df-frecs 8179  df-wrecs 8210  df-recs 8284  df-rdg 8323  df-map 8700  df-goel 33668  df-goal 33670  df-sat 33671  df-fmla 33673

This theorem is referenced by:  fmla1  33715  isfmlasuc  33716  fmlasssuc  33717  fmlaomn0  33718