Theorem satffunlem1lem2 32650

Description: Lemma 2 for satffunlem1 32654. (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 7601	. . . 4 ⊢ ∅ ∈ ω
2		satfdmfmla 32647	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
3	1, 2	mp3an3 1446	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
4		ovex 7189	. . . . . . . . . 10 ⊢ (𝑀 ↑m ω) ∈ V
5	4	difexi 5232	. . . . . . . . 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 3182	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) ∈ V)
8	4	rabex 5235	. . . . . . . . 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 3182	. . . . . . 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 514	. . . . . 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 3182	. . . . 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 5786	. . . . 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 32614	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → Rel ((𝑀 Sat 𝐸)‘∅))
16	1, 15	mp3an3 1446	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
17		1stdm 7739	. . . . . . . . . . . 12 ⊢ ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
18	16, 17	sylan 582	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
19	2	eqcomd 2827	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
20	1, 19	mp3an3 1446	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
21	20	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
22	18, 21	eleqtrrd 2916	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑢) ∈ (Fmla‘∅))
23	22	adantr 483	. . . . . . . . 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 7163	. . . . . . . . . . . . 13 ⊢ (𝑓 = (1st ‘𝑢) → (𝑓⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔𝑔))
25	24	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑓 = (1st ‘𝑢) → (𝑥 = (𝑓⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
26	25	rexbidv 3297	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑢) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔)))
27		eqidd 2822	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (1st ‘𝑢) → 𝑖 = 𝑖)
28		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (1st ‘𝑢) → 𝑓 = (1st ‘𝑢))
29	27, 28	goaleq12d 32598	. . . . . . . . . . . . 13 ⊢ (𝑓 = (1st ‘𝑢) → ∀𝑔𝑖𝑓 = ∀𝑔𝑖(1st ‘𝑢))
30	29	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑓 = (1st ‘𝑢) → (𝑥 = ∀𝑔𝑖𝑓 ↔ 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))
31	30	rexbidv 3297	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑢) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))
32	26, 31	orbi12d 915	. . . . . . . . . 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 484	. . . . . . . . 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 7739	. . . . . . . . . . . . . . . . 17 ⊢ ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
35	16, 34	sylan 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
36	20	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
37	35, 36	eleqtrrd 2916	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st ‘𝑣) ∈ (Fmla‘∅))
38	37	ad4ant13 749	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))) → (1st ‘𝑣) ∈ (Fmla‘∅))
39		oveq2 7164	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (1st ‘𝑣) → ((1st ‘𝑢)⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
40	39	eqeq2d 2832	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (1st ‘𝑣) → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
41	40	adantl 484	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 3626	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 3284	. . . . . . . . . . 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 962	. . . . . . . . . 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 409	. . . . . . . . 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 3626	. . . . . . . 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 415	. . . . . . 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 3284	. . . . . 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 7742	. . . . . . . . . 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 2898	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ 𝑓 ∈ (Fmla‘∅)))
54	52, 53	bitr3d 283	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑢) = 𝑓 ↔ 𝑓 ∈ (Fmla‘∅)))
55		r19.41v 3347	. . . . . . . . . 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 7163	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑢) = 𝑓 → ((1st ‘𝑢)⊼𝑔𝑔) = (𝑓⊼𝑔𝑔))
57	56	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑢) = 𝑓 → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = (𝑓⊼𝑔𝑔)))
58	57	rexbidv 3297	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑢) = 𝑓 → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔)))
59		eqidd 2822	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑢) = 𝑓 → 𝑖 = 𝑖)
60		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑢) = 𝑓 → (1st ‘𝑢) = 𝑓)
61	59, 60	goaleq12d 32598	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑢) = 𝑓 → ∀𝑔𝑖(1st ‘𝑢) = ∀𝑔𝑖𝑓)
62	61	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑢) = 𝑓 → (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ 𝑥 = ∀𝑔𝑖𝑓))
63	62	rexbidv 3297	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑢) = 𝑓 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
64	58, 63	orbi12d 915	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
67	66	eleq2d 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ 𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅)))
68		releldm2 7742	. . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st ‘𝑣) = 𝑔))
71		r19.41v 3347	. . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑣) = 𝑔 → ((1st ‘𝑢)⊼𝑔𝑔) = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))
73	72	eqeq2d 2832	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑣) = 𝑔 → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
74	73	biimpa 479	. . . . . . . . . . . . . . . . . . . . . 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 3273	. . . . . . . . . . . . . . . . . . . 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	syl5bir 245	. . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑔 ∈ (Fmla‘∅) → (𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)))))
80	79	rexlimdv 3283	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣))))
81	80	adantr 483	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 962	. . . . . . . . . . . . 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 262	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 3274	. . . . . . . . . 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	syl5bir 245	. . . . . . . . 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 418	. . . . . . . 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 262	. . . . . . 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 3283	. . . . . 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 214	. . . . 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 2885	. . . 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 2856	. . 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 4190	. 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 32645	. 2 ⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}) = ∅
96	94, 95	syl6eq 2872	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 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  {csn 4567  ⟨cop 4573  {copab 5128  dom cdm 5555   ↾ cres 5557  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1^st c1st 7687  2^nd c2nd 7688   ↑_m cmap 8406  ⊼_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-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-int 4877  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-1o 8102  df-2o 8103  df-map 8408  df-goel 32587  df-gona 32588  df-goal 32589  df-sat 32590  df-fmla 32592

This theorem is referenced by:  satffunlem1  32654