satfdm - Mathbox for Mario Carneiro

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Theorem satfdm 32729

Description: The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)

**Assertion**
Ref	Expression
satfdm	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))

Distinct variable groups: 𝑛,𝐸 𝑛,𝐹 𝑛,𝑀 𝑛,𝑁 𝑛,𝑉 𝑛,𝑊 𝑛,𝑋 𝑛,𝑌

Proof of Theorem satfdm Dummy variables 𝑎 𝑏 𝑖 𝑢 𝑣 𝑥 𝑦 𝑓 𝑤 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘∅))
2	1	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = ∅ → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘∅))
3		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘∅))
4	3	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = ∅ → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘∅))
5	2, 4	eqeq12d 2814	. . . . 5 ⊢ (𝑥 = ∅ → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅)))
6	5	imbi2d 344	. . . 4 ⊢ (𝑥 = ∅ → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅))))
7		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
8	7	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = 𝑦 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘𝑦))
9		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘𝑦))
10	9	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = 𝑦 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑦))
11	8, 10	eqeq12d 2814	. . . . 5 ⊢ (𝑥 = 𝑦 → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
12	11	imbi2d 344	. . . 4 ⊢ (𝑥 = 𝑦 → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦))))
13		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
14	13	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘suc 𝑦))
15		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘suc 𝑦))
16	15	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))
17	14, 16	eqeq12d 2814	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦)))
18	17	imbi2d 344	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
19		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑛))
20	19	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘𝑛))
21		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘𝑛))
22	21	dmeqd 5738	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑛))
23	20, 22	eqeq12d 2814	. . . . 5 ⊢ (𝑥 = 𝑛 → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛)))
24	23	imbi2d 344	. . . 4 ⊢ (𝑥 = 𝑛 → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))))
25		rexcom4 3212	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑦∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}))
26	25	rexbii 3210	. . . . . . . . 9 ⊢ (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑦∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}))
27		ovex 7168	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ↑_m ω) ∈ V
28	27	rabex 5199	. . . . . . . . . . . . . . 15 ⊢ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)} ∈ V
29	28	isseti 3455	. . . . . . . . . . . . . 14 ⊢ ∃𝑦 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}
30		ovex 7168	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ↑_m ω) ∈ V
31	30	rabex 5199	. . . . . . . . . . . . . . 15 ⊢ {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)} ∈ V
32	31	isseti 3455	. . . . . . . . . . . . . 14 ⊢ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}
33	29, 32	2th 267	. . . . . . . . . . . . 13 ⊢ (∃𝑦 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)} ↔ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})
34	33	anbi2i 625	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑢∈_𝑔𝑣) ∧ ∃𝑦 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ (𝑥 = (𝑢∈_𝑔𝑣) ∧ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
35		19.42v 1954	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ (𝑥 = (𝑢∈_𝑔𝑣) ∧ ∃𝑦 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}))
36		19.42v 1954	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}) ↔ (𝑥 = (𝑢∈_𝑔𝑣) ∧ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
37	34, 35, 36	3bitr4i 306	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
38	37	rexbii 3210	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
39	38	rexbii 3210	. . . . . . . . 9 ⊢ (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
40		rexcom4 3212	. . . . . . . . 9 ⊢ (∃𝑢 ∈ ω ∃𝑦∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}))
41	26, 39, 40	3bitr3ri 305	. . . . . . . 8 ⊢ (∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
42		rexcom4 3212	. . . . . . . . 9 ⊢ (∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}) ↔ ∃𝑧∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
43	42	rexbii 3210	. . . . . . . 8 ⊢ (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑧∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
44	41, 43	bitri 278	. . . . . . 7 ⊢ (∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑧∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
45		rexcom4 3212	. . . . . . 7 ⊢ (∃𝑢 ∈ ω ∃𝑧∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}) ↔ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
46	44, 45	bitri 278	. . . . . 6 ⊢ (∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)}) ↔ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)}))
47	46	abbii 2863	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})} = {𝑥 ∣ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})}
48		eqid 2798	. . . . . . . . 9 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
49	48	satfv0 32718	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})})
50	49	dmeqd 5738	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})})
51		dmopab 5748	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})} = {𝑥 ∣ ∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})}
52	50, 51	eqtrdi 2849	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = {𝑥 ∣ ∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})})
53	52	adantr 484	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = {𝑥 ∣ ∃𝑦∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑢)𝐸(𝑎‘𝑣)})})
54		eqid 2798	. . . . . . . . 9 ⊢ (𝑁 Sat 𝐹) = (𝑁 Sat 𝐹)
55	54	satfv0 32718	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → ((𝑁 Sat 𝐹)‘∅) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})})
56	55	dmeqd 5738	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → dom ((𝑁 Sat 𝐹)‘∅) = dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})})
57		dmopab 5748	. . . . . . 7 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})} = {𝑥 ∣ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})}
58	56, 57	eqtrdi 2849	. . . . . 6 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → dom ((𝑁 Sat 𝐹)‘∅) = {𝑥 ∣ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})})
59	58	adantl 485	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑁 Sat 𝐹)‘∅) = {𝑥 ∣ ∃𝑧∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢∈_𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁 ↑_m ω) ∣ (𝑎‘𝑢)𝐹(𝑎‘𝑣)})})
60	47, 53, 59	3eqtr4a 2859	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅))
61		pm2.27 42	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
62	61	adantl 485	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
63		simpr 488	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦))
64		simprl 770	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊))
65		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → 𝑦 ∈ ω)
66		df-3an 1086	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑦 ∈ ω) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑦 ∈ ω))
67	64, 65, 66	sylanbrc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑦 ∈ ω))
68		satfdmlem 32728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑦 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎))))
69	67, 68	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎))))
70		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))
71		df-3an 1086	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑦 ∈ ω) ↔ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) ∧ 𝑦 ∈ ω))
72	70, 65, 71	sylanbrc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑦 ∈ ω))
73		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦))
74	73	eqcomd 2804	. . . . . . . . . . . . . . . . 17 ⊢ (dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦) → dom ((𝑁 Sat 𝐹)‘𝑦) = dom ((𝑀 Sat 𝐸)‘𝑦))
75		satfdmlem 32728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑦 ∈ ω) ∧ dom ((𝑁 Sat 𝐹)‘𝑦) = dom ((𝑀 Sat 𝐸)‘𝑦)) → (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))))
76	72, 74, 75	syl2an 598	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))))
77	69, 76	impbid 215	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎))))
78	27	difexi 5196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) ∈ V
79	78	isseti 3455	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))
80	79	biantru 533	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ↔ (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
81	80	bicomi 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)))
82	81	rexbii 3210	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)))
83	27	rabex 5199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} ∈ V
84	83	isseti 3455	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}
85	84	biantru 533	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
86	85	bicomi 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))
87	86	rexbii 3210	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))
88	82, 87	orbi12i 912	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))
89	88	rexbii 3210	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))
90	30	difexi 5196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏))) ∈ V
91	90	isseti 3455	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))
92	91	biantru 533	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ↔ (𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
93	92	bicomi 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ 𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)))
94	93	rexbii 3210	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)))
95	30	rabex 5199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)} ∈ V
96	95	isseti 3455	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}
97	96	biantru 533	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
98	97	bicomi 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎))
99	98	rexbii 3210	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎))
100	94, 99	orbi12i 912	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎)))
101	100	rexbii 3210	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑎)))
102	77, 89, 101	3bitr4g 317	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))))
103		19.42v 1954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑤(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
104	103	bicomi 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑤(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
105	104	rexbii 3210	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
106		rexcom4 3212	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑤∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
107	105, 106	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑤∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
108		19.42v 1954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑤(𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
109	108	bicomi 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑤(𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
110	109	rexbii 3210	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω ∃𝑤(𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
111		rexcom4 3212	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ ω ∃𝑤(𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑤∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
112	110, 111	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑤∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
113	107, 112	orbi12i 912	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑤∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑤∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
114		19.43 1883	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑤∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑤∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
115	114	bicomi 227	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑤∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑤∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
116	113, 115	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
117	116	rexbii 3210	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
118		rexcom4 3212	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑤∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
119	117, 118	bitri 278	. . . . . . . . . . . . . 14 ⊢ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑤∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
120		19.42v 1954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ (𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
121	120	bicomi 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ ∃𝑧(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
122	121	rexbii 3210	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
123		rexcom4 3212	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ ∃𝑧∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
124	122, 123	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ↔ ∃𝑧∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))))
125		19.42v 1954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧(𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
126	125	bicomi 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ ∃𝑧(𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
127	126	rexbii 3210	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ ∃𝑖 ∈ ω ∃𝑧(𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
128		rexcom4 3212	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ ω ∃𝑧(𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ ∃𝑧∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
129	127, 128	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}) ↔ ∃𝑧∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))
130	124, 129	orbi12i 912	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ (∃𝑧∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑧∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
131		19.43 1883	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ (∃𝑧∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑧∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
132	131	bicomi 227	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑧∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑧∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
133	130, 132	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
134	133	rexbii 3210	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
135		rexcom4 3212	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑧∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
136	134, 135	bitri 278	. . . . . . . . . . . . . 14 ⊢ (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})) ↔ ∃𝑧∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)})))
137	102, 119, 136	3bitr3g 316	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑤∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑧∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))))
138	137	abbidv 2862	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → {𝑥 ∣ ∃𝑤∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = {𝑥 ∣ ∃𝑧∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))})
139		dmopab 5748	. . . . . . . . . . . 12 ⊢ dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = {𝑥 ∣ ∃𝑤∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}
140		dmopab 5748	. . . . . . . . . . . 12 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))} = {𝑥 ∣ ∃𝑧∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}
141	138, 139, 140	3eqtr4g 2858	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))})
142	63, 141	uneq12d 4091	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (dom ((𝑀 Sat 𝐸)‘𝑦) ∪ dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = (dom ((𝑁 Sat 𝐹)‘𝑦) ∪ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}))
143		dmun 5743	. . . . . . . . . 10 ⊢ dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = (dom ((𝑀 Sat 𝐸)‘𝑦) ∪ dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})
144		dmun 5743	. . . . . . . . . 10 ⊢ dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}) = (dom ((𝑁 Sat 𝐹)‘𝑦) ∪ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))})
145	142, 143, 144	3eqtr4g 2858	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}))
146		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉)
147	146	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → 𝑀 ∈ 𝑉)
148		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊)
149	148	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → 𝐸 ∈ 𝑊)
150	48	satfvsuc 32721	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑦 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑦) = (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
151	147, 149, 65, 150	syl2an23an 1420	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → ((𝑀 Sat 𝐸)‘suc 𝑦) = (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
152	151	dmeqd 5738	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
153		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → 𝑁 ∈ 𝑋)
154		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → 𝐹 ∈ 𝑌)
155	54	satfvsuc 32721	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌 ∧ 𝑦 ∈ ω) → ((𝑁 Sat 𝐹)‘suc 𝑦) = (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}))
156	153, 154, 65, 155	syl2an23an 1420	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → ((𝑁 Sat 𝐹)‘suc 𝑦) = (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}))
157	156	dmeqd 5738	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → dom ((𝑁 Sat 𝐹)‘suc 𝑦) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))}))
158	152, 157	eqeq12d 2814	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦) ↔ dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))})))
159	158	adantr 484	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦) ↔ dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀 ↑_m ω) ∣ ∀𝑓 ∈ 𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1^st ‘𝑎)⊼_𝑔(1^st ‘𝑏)) ∧ 𝑧 = ((𝑁 ↑_m ω) ∖ ((2^nd ‘𝑎) ∩ (2^nd ‘𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁 ↑_m ω) ∣ ∀𝑓 ∈ 𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑎)}))})))
160	145, 159	mpbird 260	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))
161	160	ex 416	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → (dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦)))
162	62, 161	syld 47	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌))) → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦)))
163	162	ex 416	. . . . 5 ⊢ (𝑦 ∈ ω → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
164	163	com23 86	. . . 4 ⊢ (𝑦 ∈ ω → ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
165	6, 12, 18, 24, 60, 164	finds 7589	. . 3 ⊢ (𝑛 ∈ ω → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛)))
166	165	impcom 411	. 2 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) ∧ 𝑛 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
167	166	ralrimiva 3149	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 {crab 3110 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 ⟨cop 4531 class class class wbr 5030 {copab 5092 dom cdm 5519 ↾ cres 5521 suc csuc 6161 ‘cfv 6324 (class class class)co 7135 ωcom 7560 1^st c1st 7669 2^nd c2nd 7670 ↑_m cmap 8389 ∈_𝑔cgoe 32693 ⊼_𝑔cgna 32694 ∀_𝑔cgol 32695 Sat csat 32696

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-goel 32700 df-goal 32702 df-sat 32703

This theorem is referenced by: satfdmfmla 32760

W3C validator