Theorem fmlasuc 35614

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 35612	. 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 2739	. . . . . . . 8 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
3	2	satf0op 35605	. . . . . . 7 ⊢ (𝑁 ∈ ω → (𝑦 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑧(𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
4		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
5	4	oveq2d 7372	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
6	5	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤))))
7	6	cbvrexvw 3218	. . . . . . . . 9 ⊢ (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)))
8	7	orbi1i 919	. . . . . . . 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 35613	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑧 ∈ (Fmla‘𝑁) ↔ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
10	9	biimprd 249	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑧 ∈ (Fmla‘𝑁)))
11	10	adantld 491	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → ((𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑧 ∈ (Fmla‘𝑁)))
12	11	imp 407	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑧 ∈ (Fmla‘𝑁))
13		vex 3435	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
14		0ex 5229	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15	13, 14	op1std 7941	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (1st ‘𝑦) = 𝑧)
16	15	eleq1d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
17	16	ad2antrl 734	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑦) ∈ (Fmla‘𝑁) ↔ 𝑧 ∈ (Fmla‘𝑁)))
18	12, 17	mpbird 258	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ω ∧ (𝑦 = ⟨𝑧, ∅⟩ ∧ ⟨𝑧, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑦) ∈ (Fmla‘𝑁))
19	18	3adant3 1138	. . . . . . . . . . 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 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → (𝑢⊼𝑔𝑣) = ((1st ‘𝑦)⊼𝑔𝑣))
21	20	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = (𝑢⊼𝑔𝑣) ↔ 𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
22	21	rexbidv 3163	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ↔ ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔𝑣)))
23		eqidd 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑖 = 𝑖)
24		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (1st ‘𝑦) → 𝑢 = (1st ‘𝑦))
25	23, 24	goaleq12d 35579	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (1st ‘𝑦) → ∀𝑔𝑖𝑢 = ∀𝑔𝑖(1st ‘𝑦))
26	25	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑦) → (𝑥 = ∀𝑔𝑖𝑢 ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
27	26	rexbidv 3163	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑦) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦)))
28	22, 27	orbi12d 924	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 35605	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) ↔ ∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
31		fmlafvel 35613	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (Fmla‘𝑁) ↔ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
32	31	biimprd 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ω → (⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁) → 𝑦 ∈ (Fmla‘𝑁)))
33	32	adantld 491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → 𝑦 ∈ (Fmla‘𝑁)))
34	33	imp 407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → 𝑦 ∈ (Fmla‘𝑁))
35		vex 3435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
36	35, 14	op1std 7941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → (1st ‘𝑤) = 𝑦)
37	36	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ⟨𝑦, ∅⟩ → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
38	37	ad2antrl 734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → ((1st ‘𝑤) ∈ (Fmla‘𝑁) ↔ 𝑦 ∈ (Fmla‘𝑁)))
39	34, 38	mpbird 258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
40	39	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → (1st ‘𝑤) ∈ (Fmla‘𝑁))
41		oveq2 7364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (1st ‘𝑤) → (𝑧⊼𝑔𝑣) = (𝑧⊼𝑔(1st ‘𝑤)))
42	41	eqeq2d 2750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (1st ‘𝑤) → (𝑥 = (𝑧⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
43	42	adantl 482	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → 𝑥 = (𝑧⊼𝑔(1st ‘𝑤)))
45	40, 43, 44	rspcedvd 3562	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ω ∧ (𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))) ∧ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))
46	45	exp31 420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ω → ((𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
47	46	exlimdv 1940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ω → (∃𝑦(𝑤 = ⟨𝑦, ∅⟩ ∧ ⟨𝑦, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
48	30, 47	sylbid 241	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (𝑤 ∈ ((∅ Sat ∅)‘𝑁) → (𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣))))
49	48	rexlimdv 3138	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤)) → ∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑧⊼𝑔𝑣)))
50	49	adantr 481	. . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) = (𝑧⊼𝑔(1st ‘𝑤)))
52	51	eqeq2d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ 𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
53	52	rexbidv 3163	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑤)) ↔ ∃𝑤 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑧⊼𝑔(1st ‘𝑤))))
54	15	oveq1d 7371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → ((1st ‘𝑦)⊼𝑔𝑣) = (𝑧⊼𝑔𝑣))
55	54	eqeq2d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔𝑣) ↔ 𝑥 = (𝑧⊼𝑔𝑣)))
56	55	rexbidv 3163	. . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . 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 258	. . . . . . . . . . . . 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 973	. . . . . . . . . . . 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 1123	. . . . . . . . . . 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 3562	. . . . . . . . . 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 1125	. . . . . . . . 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 1940	. . . . . . . 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 256	. . . . . . 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 241	. . . . . 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 3138	. . . . 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 35613	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑢 ∈ (Fmla‘𝑁) ↔ ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
69	68	biimpa 477	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
70	69	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) → ⟨𝑢, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
71		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
72	71, 14	op1std 7941	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (1st ‘𝑦) = 𝑢)
73	72	oveq1d 7371	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔(1st ‘𝑧)))
74	73	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
75	74	rexbidv 3163	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑦)⊼𝑔(1st ‘𝑧)) ↔ ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
76		eqidd 2740	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → 𝑖 = 𝑖)
77	76, 72	goaleq12d 35579	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → ∀𝑔𝑖(1st ‘𝑦) = ∀𝑔𝑖𝑢)
78	77	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ 𝑥 = ∀𝑔𝑖𝑢))
79	78	rexbidv 3163	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑢, ∅⟩ → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑦) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
80	75, 79	orbi12d 924	. . . . . . . 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 482	. . . . . . 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 35613	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) ↔ ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
83	82	biimpd 230	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
84	83	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (𝑣 ∈ (Fmla‘𝑁) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
85	84	imp 407	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
86	85	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ⟨𝑣, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
87		vex 3435	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
88	87, 14	op1std 7941	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (1st ‘𝑧) = 𝑣)
89	88	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑢⊼𝑔(1st ‘𝑧)) = (𝑢⊼𝑔𝑣))
90	89	eqeq2d 2750	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑣, ∅⟩ → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
91	90	adantl 482	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) ∧ 𝑧 = ⟨𝑣, ∅⟩) → (𝑥 = (𝑢⊼𝑔(1st ‘𝑧)) ↔ 𝑥 = (𝑢⊼𝑔𝑣)))
92		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → 𝑥 = (𝑢⊼𝑔𝑣))
93	86, 91, 92	rspcedvd 3562	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) ∧ 𝑣 ∈ (Fmla‘𝑁)) ∧ 𝑥 = (𝑢⊼𝑔𝑣)) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧)))
94	93	rexlimdva2 3142	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘𝑁)) → (∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) → ∃𝑧 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = (𝑢⊼𝑔(1st ‘𝑧))))
95	94	orim1d 973	. . . . . . . 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 407	. . . . . . 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 3562	. . . . . 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 3142	. . . . 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 213	. . . 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 2805	. . 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 4098	. 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 2774	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 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∃wrex 3063   ∪ cun 3881  ∅c0 4261  ⟨cop 4561  suc csuc 6312  ‘cfv 6485  (class class class)co 7356  ωcom 7806  1^st c1st 7929  ⊼_𝑔cgna 35562  ∀_𝑔cgol 35563   Sat csat 35564  Fmlacfmla 35565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-map 8765  df-goel 35568  df-goal 35570  df-sat 35571  df-fmla 35573

This theorem is referenced by:  fmla1  35615  isfmlasuc  35616  fmlasssuc  35617  fmlaomn0  35618