Theorem fmlafvel 32632

Description: A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)

Assertion

Ref	Expression
fmlafvel	⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))

Proof of Theorem fmlafvel

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

Step	Hyp	Ref	Expression
1		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = ∅ → (Fmla‘𝑥) = (Fmla‘∅))
2	1	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘∅)))
3		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
4	3	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = ∅ → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
5	2, 4	bibi12d 348	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅))))
6	5	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))))
7		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (Fmla‘𝑥) = (Fmla‘𝑦))
8	7	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑦)))
9		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
10	9	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
11	8, 10	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
12	11	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))))
13		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (Fmla‘𝑥) = (Fmla‘suc 𝑦))
14	13	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘suc 𝑦)))
15		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
16	15	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
17	14, 16	bibi12d 348	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
19		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (Fmla‘𝑥) = (Fmla‘𝑁))
20	19	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐹 ∈ (Fmla‘𝑥) ↔ 𝐹 ∈ (Fmla‘𝑁)))
21		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
22	21	eleq2d 2898	. . . . . 6 ⊢ (𝑥 = 𝑁 → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
23	20, 22	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥)) ↔ (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
24	23	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑥) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑥))) ↔ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))))
25		eqeq1 2825	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝐹 = (𝑖∈𝑔𝑗)))
26	25	2rexbidv 3300	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)))
27	26	elrab 3680	. . . . . 6 ⊢ (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} ↔ (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)))
28		eqidd 2822	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) → ∅ = ∅)
29		simpr 487	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))
30	28, 29	jca 514	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)))
31		simpr 487	. . . . . . . . 9 ⊢ ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))
32	31	anim2i 618	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)))
33	32	ex 415	. . . . . . 7 ⊢ (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) → (𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
34	30, 33	impbid2 228	. . . . . 6 ⊢ (𝐹 ∈ V → ((𝐹 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
35	27, 34	syl5bb 285	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
36		fmla0 32629	. . . . . . 7 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
37	36	eleq2i 2904	. . . . . 6 ⊢ (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)})
38	37	a1i 11	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ 𝐹 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}))
39		satf00 32621	. . . . . . . 8 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
40	39	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ V → ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})
41	40	eleq2d 2898	. . . . . 6 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}))
42		0ex 5211	. . . . . . 7 ⊢ ∅ ∈ V
43		eqeq1 2825	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
44	43, 26	bi2anan9r 638	. . . . . . . 8 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = ∅) → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
45	44	opelopabga 5420	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
46	42, 45	mpan2 689	. . . . . 6 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
47	41, 46	bitrd 281	. . . . 5 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝐹 = (𝑖∈𝑔𝑗))))
48	35, 38, 47	3bitr4d 313	. . . 4 ⊢ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘∅) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘∅)))
49		eqid 2821	. . . . . . . . . . . 12 ⊢ ∅ = ∅
50	49	biantrur 533	. . . . . . . . . . 11 ⊢ (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))))
51	50	bicomi 226	. . . . . . . . . 10 ⊢ ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))
52	51	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ V → ((∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))))
53		eqeq1 2825	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
54		eqeq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐹 → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ↔ 𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
55	54	rexbidv 3297	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐹 → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
56		eqeq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐹 → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))
57	56	rexbidv 3297	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐹 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))
58	55, 57	orbi12d 915	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐹 → ((∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) ↔ (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))))
59	58	rexbidv 3297	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐹 → (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))))
60	53, 59	bi2anan9r 638	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐹 ∧ 𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))))
61	60	opelopabga 5420	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ ∅ ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))))
62	42, 61	mpan2 689	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢)))))
63	59	elabg 3666	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))} ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝐹 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖(1st ‘𝑢))))
64	52, 62, 63	3bitr4d 313	. . . . . . . 8 ⊢ (𝐹 ∈ V → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
65	64	adantl 484	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))} ↔ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
66	65	orbi2d 912	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
67		eqid 2821	. . . . . . . . . 10 ⊢ (∅ Sat ∅) = (∅ Sat ∅)
68	67	satf0suc 32623	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
69	68	eleq2d 2898	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})))
70		elun 4125	. . . . . . . 8 ⊢ (⟨𝐹, ∅⟩ ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
71	69, 70	syl6bb 289	. . . . . . 7 ⊢ (𝑦 ∈ ω → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})))
72	71	ad2antrr 724	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ ⟨𝐹, ∅⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})))
73		fmlasuc0 32631	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (Fmla‘suc 𝑦) = ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
74	73	eleq2d 2898	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
75	74	ad2antrr 724	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ 𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
76		elun 4125	. . . . . . . 8 ⊢ (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
77	76	a1i 11	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ ((Fmla‘𝑦) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}) ↔ (𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
78		simpr 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))))
79	78	imp 409	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))
80	79	orbi1d 913	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → ((𝐹 ∈ (Fmla‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
81	75, 77, 80	3bitrd 307	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ (⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝐹 ∈ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})))
82	66, 72, 81	3bitr4rd 314	. . . . 5 ⊢ (((𝑦 ∈ ω ∧ (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦)))) ∧ 𝐹 ∈ V) → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))
83	82	exp31 422	. . . 4 ⊢ (𝑦 ∈ ω → ((𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑦))) → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘suc 𝑦) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘suc 𝑦)))))
84	6, 12, 18, 24, 48, 83	finds 7608	. . 3 ⊢ (𝑁 ∈ ω → (𝐹 ∈ V → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
85	84	com12 32	. 2 ⊢ (𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
86		prcnel 3518	. . . . 5 ⊢ (¬ 𝐹 ∈ V → ¬ 𝐹 ∈ (Fmla‘𝑁))
87	86	adantr 483	. . . 4 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ 𝐹 ∈ (Fmla‘𝑁))
88		opprc1 4827	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → ⟨𝐹, ∅⟩ = ∅)
89	88	adantr 483	. . . . 5 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ⟨𝐹, ∅⟩ = ∅)
90		satf0n0 32625	. . . . . . 7 ⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
91		df-nel 3124	. . . . . . 7 ⊢ (∅ ∉ ((∅ Sat ∅)‘𝑁) ↔ ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
92	90, 91	sylib 220	. . . . . 6 ⊢ (𝑁 ∈ ω → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
93	92	adantl 484	. . . . 5 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ∅ ∈ ((∅ Sat ∅)‘𝑁))
94	89, 93	eqneltrd 2932	. . . 4 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → ¬ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))
95	87, 94	2falsed 379	. . 3 ⊢ ((¬ 𝐹 ∈ V ∧ 𝑁 ∈ ω) → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
96	95	ex 415	. 2 ⊢ (¬ 𝐹 ∈ V → (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁))))
97	85, 96	pm2.61i 184	1 ⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  {cab 2799   ∉ wnel 3123  ∃wrex 3139  {crab 3142  Vcvv 3494   ∪ cun 3934  ∅c0 4291  ⟨cop 4573  {copab 5128  suc csuc 6193  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1^st c1st 7687  ∈_𝑔cgoe 32580  ⊼_𝑔cgna 32581  ∀_𝑔cgol 32582   Sat csat 32583  Fmlacfmla 32584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-map 8408  df-goel 32587  df-sat 32590  df-fmla 32592

This theorem is referenced by:  fmlasuc  32633