Theorem satffunlem1lem2 35408

Description: Lemma 2 for satffunlem1 35412. (Contributed by AV, 23-Oct-2023.)

Assertion

Ref	Expression
satffunlem1lem2	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = ∅)

Distinct variable groups:   𝑓,𝐸,𝑖,𝑢,𝑣,𝑥,𝑦   𝑓,𝑀,𝑖,𝑢,𝑣,𝑥,𝑦   𝑓,𝑉,𝑖,𝑢,𝑣,𝑥   𝑓,𝑊,𝑖,𝑢,𝑣,𝑥   𝑥,𝑗,𝑦

Allowed substitution hints:   𝐸(𝑗)   𝑀(𝑗)   𝑉(𝑦,𝑗)   𝑊(𝑦,𝑗)

Proof of Theorem satffunlem1lem2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7910	. . . 4 ⊢ ∅ ∈ ω
2		satfdmfmla 35405	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
3	1, 2	mp3an3 1452	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
4		ovex 7464	. . . . . . . . . 10 ⊢ (𝑀 ↑m ω) ∈ V
5	4	difexi 5330	. . . . . . . . 9 ⊢ ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V
6	5	a1i 11	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V)
7	6	ralrimiva 3146	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V)
8	4	rabex 5339	. . . . . . . . 9 ⊢ {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑖 ∈ ω) → {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V)
10	9	ralrimiva 3146	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑖 ∈ ω {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V)
11	7, 10	jca 511	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V))
12	11	ralrimiva 3146	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ∀𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V))
13		dmopab2rex 5928	. . . . 5 ⊢ (∀𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ∈ V) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} = {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})
14	12, 13	syl 17	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} = {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))})
15		satfrel 35372	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → Rel ((𝑀 Sat 𝐸)‘∅))
16	1, 15	mp3an3 1452	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
17		1stdm 8065	. . . . . . . . . . . 12 ⊢ ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
18	16, 17	sylan 580	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
19	2	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
20	1, 19	mp3an3 1452	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
21	20	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
22	18, 21	eleqtrrd 2844	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ (Fmla‘∅))
23	22	adantr 480	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))) → (1st ‘𝑢) ∈ (Fmla‘∅))
24		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑓 = (1st ‘𝑢) → (𝑓⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔𝑔))
25	24	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑓 = (1st ‘𝑢) → (𝑥 = (𝑓⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
26	25	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑢) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
27		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (1st ‘𝑢) → 𝑖 = 𝑖)
28		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (1st ‘𝑢) → 𝑓 = (1st ‘𝑢))
29	27, 28	goaleq12d 35356	. . . . . . . . . . . . 13 ⊢ (𝑓 = (1st ‘𝑢) → ∀𝑔𝑖𝑓 = ∀𝑔𝑖(1st ‘𝑢))
30	29	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑓 = (1st ‘𝑢) → (𝑥 = ∀𝑔𝑖𝑓 ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))
31	30	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑢) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))
32	26, 31	orbi12d 919	. . . . . . . . . 10 ⊢ (𝑓 = (1st ‘𝑢) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
33	32	adantl 481	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))) ∧ 𝑓 = (1st ‘𝑢)) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
34		1stdm 8065	. . . . . . . . . . . . . . . . 17 ⊢ ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
35	16, 34	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
36	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
37	35, 36	eleqtrrd 2844	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ (Fmla‘∅))
38	37	ad4ant13 751	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))) → (1st ‘𝑣) ∈ (Fmla‘∅))
39		oveq2 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (1st ‘𝑣) → ((1st ‘𝑢)⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
40	39	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (1st ‘𝑣) → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
41	40	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))) ∧ 𝑔 = (1st ‘𝑣)) → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
42		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))) → 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
43	38, 41, 42	rspcedvd 3624	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔))
44	43	ex 412	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
45	44	rexlimdva 3155	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
46	45	orim1d 968	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
47	46	imp 406	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))
48	23, 33, 47	rspcedvd 3624	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
49	48	ex 412	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
50	49	rexlimdva 3155	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
51		releldm2 8068	. . . . . . . . . 10 ⊢ (Rel ((𝑀 Sat 𝐸)‘∅) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓))
52	16, 51	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓))
53	3	eleq2d 2827	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ 𝑓 ∈ (Fmla‘∅)))
54	52, 53	bitr3d 281	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓 ↔ 𝑓 ∈ (Fmla‘∅)))
55		r19.41v 3189	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)((1st ‘𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) ↔ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
56		oveq1 7438	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑢) = 𝑓 → ((1st ‘𝑢)⊼𝑔𝑔) = (𝑓⊼𝑔𝑔))
57	56	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑢) = 𝑓 → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = (𝑓⊼𝑔𝑔)))
58	57	rexbidv 3179	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑢) = 𝑓 → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔)))
59		eqidd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑢) = 𝑓 → 𝑖 = 𝑖)
60		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑢) = 𝑓 → (1st ‘𝑢) = 𝑓)
61	59, 60	goaleq12d 35356	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑢) = 𝑓 → ∀𝑔𝑖(1st ‘𝑢) = ∀𝑔𝑖𝑓)
62	61	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑢) = 𝑓 → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ 𝑥 = ∀𝑔𝑖𝑓))
63	62	rexbidv 3179	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑢) = 𝑓 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
64	58, 63	orbi12d 919	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
65	64	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st ‘𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
66	3	eqcomd 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
67	66	eleq2d 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ 𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅)))
68		releldm2 8068	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel ((𝑀 Sat 𝐸)‘∅) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔))
69	16, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔))
70	67, 69	bitrd 279	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔))
71		r19.41v 3189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
72	39	eqcoms 2745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑣) = 𝑔 → ((1st ‘𝑢)⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
73	72	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑣) = 𝑔 → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
74	73	biimpa 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
75	74	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
76	75	reximdv 3170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
77	71, 76	biimtrrid 243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔 ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
78	77	expd 415	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔 → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))))
79	70, 78	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))))
80	79	rexlimdv 3153	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
81	80	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st ‘𝑢) = 𝑓) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
83	82	orim1d 968	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st ‘𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
84	65, 83	sylbird 260	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st ‘𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
85	84	expimpd 453	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (((1st ‘𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
86	85	reximdva 3168	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)((1st ‘𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
87	55, 86	biimtrrid 243	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
88	87	expd 415	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))))
89	54, 88	sylbird 260	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑓 ∈ (Fmla‘∅) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))))
90	89	rexlimdv 3153	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))))
91	50, 90	impbid 212	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) ↔ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
92	91	abbidv 2808	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))} = {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)})
93	14, 92	eqtrd 2777	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} = {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)})
94	3, 93	ineq12d 4221	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}))
95		fmla0disjsuc 35403	. 2 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}) = ∅
96	94, 95	eqtrdi 2793	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950  ∅c0 4333  {csn 4626  ⟨cop 4632  {copab 5205  dom cdm 5685   ↾ cres 5687  Rel wrel 5690  ‘cfv 6561  (class class class)co 7431  ωcom 7887  1^st c1st 8012  2^nd c2nd 8013   ↑_m cmap 8866  ⊼_𝑔cgna 35339  ∀_𝑔cgol 35340   Sat csat 35341  Fmlacfmla 35342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-fmla 35350

This theorem is referenced by:  satffunlem1  35412