satfv1fvfmla1 - Mathbox for Mario Carneiro

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Theorem satfv1fvfmla1 33385

Description: The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)

**Hypothesis**
Ref	Expression
satfv1fvfmla1.x	⊢ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿))

**Assertion**
Ref	Expression
satfv1fvfmla1	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})

Distinct variable groups: 𝐸,𝑎 𝐼,𝑎 𝐽,𝑎 𝐾,𝑎 𝐿,𝑎 𝑀,𝑎 Allowed substitution hints: 𝑉(𝑎) 𝑊(𝑎) 𝑋(𝑎)

Proof of Theorem satfv1fvfmla1 Dummy variables 𝑖 𝑗 𝑘 𝑛 𝑥 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉)
2		simpr 485	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊)
3		1onn 8470	. . . . . 6 ⊢ 1_o ∈ ω
4	3	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 1_o ∈ ω)
5	1, 2, 4	3jca 1127	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 1_o ∈ ω))
6	5	3ad2ant1 1132	. . 3 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 1_o ∈ ω))
7		satffun 33371	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 1_o ∈ ω) → Fun ((𝑀 Sat 𝐸)‘1_o))
8	6, 7	syl 17	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat 𝐸)‘1_o))
9		simp2l 1198	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
10		simp2r 1199	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
11		simp3l 1200	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
12		simp3r 1201	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
13		satfv1fvfmla1.x	. . . . . . . . . . 11 ⊢ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿))
14		eqid 2738	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}
15	13, 14	pm3.2i 471	. . . . . . . . . 10 ⊢ (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})
16	15	a1i 11	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}))
17		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (𝑘∈_𝑔𝑙) = (𝐾∈_𝑔𝑙))
18	17	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙)))
19	18	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ↔ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙))))
20		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐾 → (𝑎‘𝑘) = (𝑎‘𝐾))
21	20	breq1d 5084	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐾 → ((𝑎‘𝑘)𝐸(𝑎‘𝑙) ↔ (𝑎‘𝐾)𝐸(𝑎‘𝑙)))
22	21	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙) ↔ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙)))
23	22	orbi2d 913	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → ((¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙)) ↔ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))))
24	23	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))})
25	24	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))}))
26	19, 25	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))})))
27		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝐿 → (𝐾∈_𝑔𝑙) = (𝐾∈_𝑔𝐿))
28	27	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐿 → ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙)) = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿)))
29	28	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐿 → (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙)) ↔ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿))))
30		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐿 → (𝑎‘𝑙) = (𝑎‘𝐿))
31	30	breq2d 5086	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐿 → ((𝑎‘𝐾)𝐸(𝑎‘𝑙) ↔ (𝑎‘𝐾)𝐸(𝑎‘𝐿)))
32	31	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝐿 → (¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙) ↔ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿)))
33	32	orbi2d 913	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝐿 → ((¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙)) ↔ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))))
34	33	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐿 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})
35	34	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐿 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}))
36	29, 35	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑙 = 𝐿 → ((𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝑙))}) ↔ (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})))
37	26, 36	rspc2ev 3572	. . . . . . . . 9 ⊢ ((𝐾 ∈ ω ∧ 𝐿 ∈ ω ∧ (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}))
38	11, 12, 16, 37	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}))
39	38	orcd 870	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})))
40		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (𝑖∈_𝑔𝑗) = (𝐼∈_𝑔𝑗))
41	40	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)))
42	41	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ↔ 𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙))))
43		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐼 → (𝑎‘𝑖) = (𝑎‘𝐼))
44	43	breq1d 5084	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐼 → ((𝑎‘𝑖)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝑗)))
45	44	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ↔ ¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗)))
46	45	orbi1d 914	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ((¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙)) ↔ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))))
47	46	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})
48	47	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}))
49	42, 48	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
50	49	2rexbidv 3229	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
51		eqidd 2739	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → 𝑛 = 𝑛)
52	51, 40	goaleq12d 33313	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → ∀_𝑔𝑛(𝑖∈_𝑔𝑗) = ∀_𝑔𝑛(𝐼∈_𝑔𝑗))
53	52	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ↔ 𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗)))
54		eqeq1 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑛 ↔ 𝐼 = 𝑛))
55		biidd 261	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)) ↔ if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗))))
56	43	breq1d 5084	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐼 → ((𝑎‘𝑖)𝐸𝑧 ↔ (𝑎‘𝐼)𝐸𝑧))
57	56, 44	ifpbi23d 1079	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)) ↔ if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗))))
58	54, 55, 57	ifpbi123d 1077	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗))) ↔ if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))))
59	58	ralbidv 3112	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗))) ↔ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))))
60	59	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))})
61	60	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))}))
62	53, 61	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}) ↔ (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))})))
63	62	rexbidv 3226	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))})))
64	50, 63	orbi12d 916	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))}))))
65		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐼∈_𝑔𝑗) = (𝐼∈_𝑔𝐽))
66	65	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)))
67	66	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ↔ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙))))
68		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑎‘𝑗) = (𝑎‘𝐽))
69	68	breq2d 5086	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑎‘𝐼)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
70	69	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ↔ ¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽)))
71	70	orbi1d 914	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → ((¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙)) ↔ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))))
72	71	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})
73	72	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}))
74	67, 73	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
75	74	2rexbidv 3229	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
76		eqidd 2739	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → 𝑛 = 𝑛)
77	76, 65	goaleq12d 33313	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ∀_𝑔𝑛(𝐼∈_𝑔𝑗) = ∀_𝑔𝑛(𝐼∈_𝑔𝐽))
78	77	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ↔ 𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽)))
79		eqeq1 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑗 = 𝑛 ↔ 𝐽 = 𝑛))
80		biidd 261	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑧𝐸𝑧 ↔ 𝑧𝐸𝑧))
81	68	breq2d 5086	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑧𝐸(𝑎‘𝑗) ↔ 𝑧𝐸(𝑎‘𝐽)))
82	79, 80, 81	ifpbi123d 1077	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)) ↔ if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽))))
83		biidd 261	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝑎‘𝐼)𝐸𝑧 ↔ (𝑎‘𝐼)𝐸𝑧))
84	79, 83, 69	ifpbi123d 1077	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)) ↔ if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽))))
85	82, 84	ifpbi23d 1079	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗))) ↔ if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
86	85	ralbidv 3112	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗))) ↔ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))))
87	86	rabbidv 3414	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})
88	87	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ({𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))}))
89	78, 88	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))}) ↔ (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})))
90	89	rexbidv 3226	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})))
91	75, 90	orbi12d 916	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))}))))
92	64, 91	rspc2ev 3572	. . . . . . 7 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝐼∈_𝑔𝐽) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑛, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))}))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})))
93	9, 10, 39, 92	syl3anc 1370	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})))
94	13	ovexi 7309	. . . . . . . 8 ⊢ 𝑋 ∈ V
95	94	a1i 11	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ V)
96		ovex 7308	. . . . . . . 8 ⊢ (𝑀 ↑_m ω) ∈ V
97	96	rabex 5256	. . . . . . 7 ⊢ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} ∈ V
98		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ↔ 𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙))))
99		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} → (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}))
100	98, 99	bi2anan9 636	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → ((𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
101	100	2rexbidv 3229	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))})))
102		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ↔ 𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗)))
103		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} → (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))
104	102, 103	bi2anan9 636	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → ((𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}) ↔ (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})))
105	104	rexbidv 3226	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → (∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})))
106	101, 105	orbi12d 916	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))))
107	106	2rexbidv 3229	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))})) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))))
108	107	opelopabga 5446	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} ∈ V) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))))
109	95, 97, 108	sylancl 586	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))))
110	93, 109	mpbird 256	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))})
111	110	olcd 871	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘∅) ∨ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
112		elun 4083	. . . 4 ⊢ (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}) ↔ (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘∅) ∨ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
113	111, 112	sylibr 233	. . 3 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
114		eqid 2738	. . . . . 6 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
115	114	satfv1 33325	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘1_o) = (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
116	115	eleq2d 2824	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1_o) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))})))
117	116	3ad2ant1 1132	. . 3 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1_o) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈_𝑔𝑗)⊼_𝑔(𝑘∈_𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀_𝑔𝑛(𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))})))
118	113, 117	mpbird 256	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1_o))
119		funopfv 6821	. 2 ⊢ (Fun ((𝑀 Sat 𝐸)‘1_o) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1_o) → (((𝑀 Sat 𝐸)‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))}))
120	8, 118, 119	sylc 65	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 if-wif 1060 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ∪ cun 3885 ∅c0 4256 ⟨cop 4567 class class class wbr 5074 {copab 5136 Fun wfun 6427 ‘cfv 6433 (class class class)co 7275 ωcom 7712 1_oc1o 8290 ↑_m cmap 8615 ∈_𝑔cgoe 33295 ⊼_𝑔cgna 33296 ∀_𝑔cgol 33297 Sat csat 33298

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-map 8617 df-goel 33302 df-gona 33303 df-goal 33304 df-sat 33305 df-fmla 33307

This theorem is referenced by: satefvfmla1 33387

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