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Theorem satffunlem1lem1 32709
Description: Lemma for satffunlem1 32714. (Contributed by AV, 17-Oct-2023.)
Assertion
Ref Expression
satffunlem1lem1 (Fun ((𝑀 Sat 𝐸)‘𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
Distinct variable groups:   𝑖,𝐸,𝑢,𝑥,𝑦   𝑣,𝐸,𝑢,𝑥,𝑦   𝑖,𝑀,𝑢,𝑥,𝑦   𝑣,𝑀   𝑖,𝑁,𝑢,𝑥,𝑦   𝑣,𝑁   𝑓,𝑖,𝑢,𝑦   𝑣,𝑓   𝑖,𝑘,𝑢,𝑦   𝑣,𝑘
Allowed substitution hints:   𝐸(𝑓,𝑘)   𝑀(𝑓,𝑘)   𝑁(𝑓,𝑘)

Proof of Theorem satffunlem1lem1
Dummy variables 𝑗 𝑠 𝑧 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6661 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → (1st𝑢) = (1st𝑠))
2 fveq2 6661 . . . . . . . . . . . . 13 (𝑣 = 𝑟 → (1st𝑣) = (1st𝑟))
31, 2oveqan12d 7168 . . . . . . . . . . . 12 ((𝑢 = 𝑠𝑣 = 𝑟) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)))
43eqeq2d 2835 . . . . . . . . . . 11 ((𝑢 = 𝑠𝑣 = 𝑟) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))))
5 fveq2 6661 . . . . . . . . . . . . . 14 (𝑢 = 𝑠 → (2nd𝑢) = (2nd𝑠))
6 fveq2 6661 . . . . . . . . . . . . . 14 (𝑣 = 𝑟 → (2nd𝑣) = (2nd𝑟))
75, 6ineqan12d 4176 . . . . . . . . . . . . 13 ((𝑢 = 𝑠𝑣 = 𝑟) → ((2nd𝑢) ∩ (2nd𝑣)) = ((2nd𝑠) ∩ (2nd𝑟)))
87difeq2d 4085 . . . . . . . . . . . 12 ((𝑢 = 𝑠𝑣 = 𝑟) → ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))))
98eqeq2d 2835 . . . . . . . . . . 11 ((𝑢 = 𝑠𝑣 = 𝑟) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))))
104, 9anbi12d 633 . . . . . . . . . 10 ((𝑢 = 𝑠𝑣 = 𝑟) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))))))
1110cbvrexdva 3445 . . . . . . . . 9 (𝑢 = 𝑠 → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))))))
12 simpr 488 . . . . . . . . . . . . 13 ((𝑢 = 𝑠𝑖 = 𝑗) → 𝑖 = 𝑗)
131adantr 484 . . . . . . . . . . . . 13 ((𝑢 = 𝑠𝑖 = 𝑗) → (1st𝑢) = (1st𝑠))
1412, 13goaleq12d 32658 . . . . . . . . . . . 12 ((𝑢 = 𝑠𝑖 = 𝑗) → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑗(1st𝑠))
1514eqeq2d 2835 . . . . . . . . . . 11 ((𝑢 = 𝑠𝑖 = 𝑗) → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑗(1st𝑠)))
16 opeq1 4787 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → ⟨𝑖, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
1716sneqd 4562 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → {⟨𝑖, 𝑘⟩} = {⟨𝑗, 𝑘⟩})
18 sneq 4560 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → {𝑖} = {𝑗})
1918difeq2d 4085 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → (ω ∖ {𝑖}) = (ω ∖ {𝑗}))
2019reseq2d 5840 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑓 ↾ (ω ∖ {𝑖})) = (𝑓 ↾ (ω ∖ {𝑗})))
2117, 20uneq12d 4126 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) = ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))))
2221adantl 485 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑠𝑖 = 𝑗) → ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) = ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))))
235adantr 484 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑠𝑖 = 𝑗) → (2nd𝑢) = (2nd𝑠))
2422, 23eleq12d 2910 . . . . . . . . . . . . . 14 ((𝑢 = 𝑠𝑖 = 𝑗) → (({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢) ↔ ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)))
2524ralbidv 3192 . . . . . . . . . . . . 13 ((𝑢 = 𝑠𝑖 = 𝑗) → (∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢) ↔ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)))
2625rabbidv 3465 . . . . . . . . . . . 12 ((𝑢 = 𝑠𝑖 = 𝑗) → {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})
2726eqeq2d 2835 . . . . . . . . . . 11 ((𝑢 = 𝑠𝑖 = 𝑗) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}))
2815, 27anbi12d 633 . . . . . . . . . 10 ((𝑢 = 𝑠𝑖 = 𝑗) → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})))
2928cbvrexdva 3445 . . . . . . . . 9 (𝑢 = 𝑠 → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})))
3011, 29orbi12d 916 . . . . . . . 8 (𝑢 = 𝑠 → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∨ ∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}))))
3130cbvrexvw 3435 . . . . . . 7 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∨ ∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})))
32 simp-4l 782 . . . . . . . . . . . . . . 15 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → Fun ((𝑀 Sat 𝐸)‘𝑁))
33 simpr 488 . . . . . . . . . . . . . . . 16 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁))
3433anim1i 617 . . . . . . . . . . . . . . 15 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)))
35 simpr 488 . . . . . . . . . . . . . . . . 17 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁))
3635anim1i 617 . . . . . . . . . . . . . . . 16 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)))
3736ad2antrr 725 . . . . . . . . . . . . . . 15 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)))
38 satffunlem 32708 . . . . . . . . . . . . . . . . 17 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))))) → 𝑧 = 𝑦)
3938eqcomd 2830 . . . . . . . . . . . . . . . 16 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))))) → 𝑦 = 𝑧)
40393exp 1116 . . . . . . . . . . . . . . 15 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁))) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
4132, 34, 37, 40syl3anc 1368 . . . . . . . . . . . . . 14 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
4241rexlimdva 3276 . . . . . . . . . . . . 13 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
43 eqeq1 2828 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∀𝑔𝑖(1st𝑢) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ∀𝑔𝑖(1st𝑢) = ((1st𝑠)⊼𝑔(1st𝑟))))
44 df-goal 32649 . . . . . . . . . . . . . . . . . . . 20 𝑔𝑖(1st𝑢) = ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩
45 fvex 6674 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑠) ∈ V
46 fvex 6674 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑟) ∈ V
47 gonafv 32657 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑠) ∈ V ∧ (1st𝑟) ∈ V) → ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
4845, 46, 47mp2an 691 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩
4944, 48eqeq12i 2839 . . . . . . . . . . . . . . . . . . 19 (∀𝑔𝑖(1st𝑢) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
50 2oex 8108 . . . . . . . . . . . . . . . . . . . . 21 2o ∈ V
51 opex 5343 . . . . . . . . . . . . . . . . . . . . 21 𝑖, (1st𝑢)⟩ ∈ V
5250, 51opth 5355 . . . . . . . . . . . . . . . . . . . 20 (⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ ↔ (2o = 1o ∧ ⟨𝑖, (1st𝑢)⟩ = ⟨(1st𝑠), (1st𝑟)⟩))
53 1one2o 8265 . . . . . . . . . . . . . . . . . . . . . . 23 1o ≠ 2o
54 df-ne 3015 . . . . . . . . . . . . . . . . . . . . . . . 24 (1o ≠ 2o ↔ ¬ 1o = 2o)
55 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . 24 (¬ 1o = 2o → (1o = 2o → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧)))
5654, 55sylbi 220 . . . . . . . . . . . . . . . . . . . . . . 23 (1o ≠ 2o → (1o = 2o → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧)))
5753, 56ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (1o = 2o → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧))
5857eqcoms 2832 . . . . . . . . . . . . . . . . . . . . 21 (2o = 1o → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧))
5958adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((2o = 1o ∧ ⟨𝑖, (1st𝑢)⟩ = ⟨(1st𝑠), (1st𝑟)⟩) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧))
6052, 59sylbi 220 . . . . . . . . . . . . . . . . . . 19 (⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧))
6149, 60sylbi 220 . . . . . . . . . . . . . . . . . 18 (∀𝑔𝑖(1st𝑢) = ((1st𝑠)⊼𝑔(1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧))
6243, 61syl6bi 256 . . . . . . . . . . . . . . . . 17 (𝑥 = ∀𝑔𝑖(1st𝑢) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → 𝑦 = 𝑧)))
6362impd 414 . . . . . . . . . . . . . . . 16 (𝑥 = ∀𝑔𝑖(1st𝑢) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧))
6463adantr 484 . . . . . . . . . . . . . . 15 ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧))
6564a1i 11 . . . . . . . . . . . . . 14 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
6665rexlimdva 3276 . . . . . . . . . . . . 13 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
6742, 66jaod 856 . . . . . . . . . . . 12 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
6867rexlimdva 3276 . . . . . . . . . . 11 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → 𝑦 = 𝑧)))
6968com23 86 . . . . . . . . . 10 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
7069rexlimdva 3276 . . . . . . . . 9 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
71 eqeq1 2828 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ∀𝑔𝑗(1st𝑠) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∀𝑔𝑗(1st𝑠) = ((1st𝑢)⊼𝑔(1st𝑣))))
72 df-goal 32649 . . . . . . . . . . . . . . . . . . . . 21 𝑔𝑗(1st𝑠) = ⟨2o, ⟨𝑗, (1st𝑠)⟩⟩
73 fvex 6674 . . . . . . . . . . . . . . . . . . . . . 22 (1st𝑢) ∈ V
74 fvex 6674 . . . . . . . . . . . . . . . . . . . . . 22 (1st𝑣) ∈ V
75 gonafv 32657 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ V ∧ (1st𝑣) ∈ V) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
7673, 74, 75mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩
7772, 76eqeq12i 2839 . . . . . . . . . . . . . . . . . . . 20 (∀𝑔𝑗(1st𝑠) = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ⟨2o, ⟨𝑗, (1st𝑠)⟩⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
78 opex 5343 . . . . . . . . . . . . . . . . . . . . . 22 𝑗, (1st𝑠)⟩ ∈ V
7950, 78opth 5355 . . . . . . . . . . . . . . . . . . . . 21 (⟨2o, ⟨𝑗, (1st𝑠)⟩⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ↔ (2o = 1o ∧ ⟨𝑗, (1st𝑠)⟩ = ⟨(1st𝑢), (1st𝑣)⟩))
80 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . 25 (¬ 1o = 2o → (1o = 2o𝑦 = 𝑧))
8154, 80sylbi 220 . . . . . . . . . . . . . . . . . . . . . . . 24 (1o ≠ 2o → (1o = 2o𝑦 = 𝑧))
8253, 81ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (1o = 2o𝑦 = 𝑧)
8382eqcoms 2832 . . . . . . . . . . . . . . . . . . . . . 22 (2o = 1o𝑦 = 𝑧)
8483adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((2o = 1o ∧ ⟨𝑗, (1st𝑠)⟩ = ⟨(1st𝑢), (1st𝑣)⟩) → 𝑦 = 𝑧)
8579, 84sylbi 220 . . . . . . . . . . . . . . . . . . . 20 (⟨2o, ⟨𝑗, (1st𝑠)⟩⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ → 𝑦 = 𝑧)
8677, 85sylbi 220 . . . . . . . . . . . . . . . . . . 19 (∀𝑔𝑗(1st𝑠) = ((1st𝑢)⊼𝑔(1st𝑣)) → 𝑦 = 𝑧)
8771, 86syl6bi 256 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∀𝑔𝑗(1st𝑠) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → 𝑦 = 𝑧))
8887adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → 𝑦 = 𝑧))
8988com12 32 . . . . . . . . . . . . . . . 16 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧))
9089adantr 484 . . . . . . . . . . . . . . 15 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧))
9190a1i 11 . . . . . . . . . . . . . 14 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
9291rexlimdva 3276 . . . . . . . . . . . . 13 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
93 eqeq1 2828 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ∀𝑔𝑖(1st𝑢) → (𝑥 = ∀𝑔𝑗(1st𝑠) ↔ ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑗(1st𝑠)))
9444, 72eqeq12i 2839 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑔𝑖(1st𝑢) = ∀𝑔𝑗(1st𝑠) ↔ ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ = ⟨2o, ⟨𝑗, (1st𝑠)⟩⟩)
9550, 51opth 5355 . . . . . . . . . . . . . . . . . . . . . 22 (⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ = ⟨2o, ⟨𝑗, (1st𝑠)⟩⟩ ↔ (2o = 2o ∧ ⟨𝑖, (1st𝑢)⟩ = ⟨𝑗, (1st𝑠)⟩))
96 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ V
9796, 73opth 5355 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑖, (1st𝑢)⟩ = ⟨𝑗, (1st𝑠)⟩ ↔ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)))
9897anbi2i 625 . . . . . . . . . . . . . . . . . . . . . 22 ((2o = 2o ∧ ⟨𝑖, (1st𝑢)⟩ = ⟨𝑗, (1st𝑠)⟩) ↔ (2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠))))
9994, 95, 983bitri 300 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑔𝑖(1st𝑢) = ∀𝑔𝑗(1st𝑠) ↔ (2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠))))
10093, 99syl6bb 290 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ∀𝑔𝑖(1st𝑢) → (𝑥 = ∀𝑔𝑗(1st𝑠) ↔ (2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)))))
101100adantl 485 . . . . . . . . . . . . . . . . . . 19 ((((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) ∧ 𝑥 = ∀𝑔𝑖(1st𝑢)) → (𝑥 = ∀𝑔𝑗(1st𝑠) ↔ (2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)))))
102 funfv1st2nd 7740 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠))
103102ex 416 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) → (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠)))
104 funfv1st2nd 7740 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢))
105104ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢)))
106 fveqeq2 6670 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((1st𝑢) = (1st𝑠) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢) ↔ (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑢)))
107 eqtr2 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑢) ∧ (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠)) → (2nd𝑢) = (2nd𝑠))
108 opeq1 4787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑗 = 𝑖 → ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑘⟩)
109108sneqd 4562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑗 = 𝑖 → {⟨𝑗, 𝑘⟩} = {⟨𝑖, 𝑘⟩})
110 sneq 4560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑗 = 𝑖 → {𝑗} = {𝑖})
111110difeq2d 4085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑗 = 𝑖 → (ω ∖ {𝑗}) = (ω ∖ {𝑖}))
112111reseq2d 5840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑗 = 𝑖 → (𝑓 ↾ (ω ∖ {𝑗})) = (𝑓 ↾ (ω ∖ {𝑖})))
113109, 112uneq12d 4126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑗 = 𝑖 → ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) = ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))))
114113eqcoms 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑖 = 𝑗 → ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) = ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))))
115114adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) = ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))))
116 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → (2nd𝑢) = (2nd𝑠))
117116eqcomd 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → (2nd𝑠) = (2nd𝑢))
118115, 117eleq12d 2910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → (({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠) ↔ ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)))
119118ralbidv 3192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → (∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠) ↔ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)))
120119rabbidv 3465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})
121 eqeq12 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → (𝑦 = 𝑧 ↔ {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
122120, 121syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((2nd𝑢) = (2nd𝑠) ∧ 𝑖 = 𝑗) → ((𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → 𝑦 = 𝑧))
123122exp4b 434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((2nd𝑢) = (2nd𝑠) → (𝑖 = 𝑗 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
124107, 123syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑢) ∧ (((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠)) → (𝑖 = 𝑗 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
125124ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑢) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → (𝑖 = 𝑗 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))))
126106, 125syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((1st𝑢) = (1st𝑠) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → (𝑖 = 𝑗 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))))
127126com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((1st𝑢) = (1st𝑠) → (𝑖 = 𝑗 → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))))
128127impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))))
129128com13 88 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑢)) = (2nd𝑢) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))))
130105, 129syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))))
131130com23 86 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Fun ((𝑀 Sat 𝐸)‘𝑁) → ((((𝑀 Sat 𝐸)‘𝑁)‘(1st𝑠)) = (2nd𝑠) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))))
132103, 131syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))))
133132imp 410 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))))
134133adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) → (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))))
135134imp 410 . . . . . . . . . . . . . . . . . . . . 21 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
136135adantld 494 . . . . . . . . . . . . . . . . . . . 20 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠))) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
137136ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) ∧ 𝑥 = ∀𝑔𝑖(1st𝑢)) → ((2o = 2o ∧ (𝑖 = 𝑗 ∧ (1st𝑢) = (1st𝑠))) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
138101, 137sylbid 243 . . . . . . . . . . . . . . . . . 18 ((((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) ∧ 𝑥 = ∀𝑔𝑖(1st𝑢)) → (𝑥 = ∀𝑔𝑗(1st𝑠) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
139138impd 414 . . . . . . . . . . . . . . . . 17 ((((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) ∧ 𝑥 = ∀𝑔𝑖(1st𝑢)) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧)))
140139ex 416 . . . . . . . . . . . . . . . 16 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) → (𝑥 = ∀𝑔𝑖(1st𝑢) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 = 𝑧))))
141140com34 91 . . . . . . . . . . . . . . 15 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) → (𝑥 = ∀𝑔𝑖(1st𝑢) → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧))))
142141impd 414 . . . . . . . . . . . . . 14 (((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑖 ∈ ω) → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
143142rexlimdva 3276 . . . . . . . . . . . . 13 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
14492, 143jaod 856 . . . . . . . . . . . 12 ((((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
145144rexlimdva 3276 . . . . . . . . . . 11 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → 𝑦 = 𝑧)))
146145com23 86 . . . . . . . . . 10 (((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) ∧ 𝑗 ∈ ω) → ((𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
147146rexlimdva 3276 . . . . . . . . 9 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → (∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)}) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
14870, 147jaod 856 . . . . . . . 8 ((Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ 𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)) → ((∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∨ ∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
149148rexlimdva 3276 . . . . . . 7 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (∃𝑠 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑟 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∨ ∃𝑗 ∈ ω (𝑥 = ∀𝑔𝑗(1st𝑠) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑗, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑗}))) ∈ (2nd𝑠)})) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
15031, 149syl5bi 245 . . . . . 6 (Fun ((𝑀 Sat 𝐸)‘𝑁) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑦 = 𝑧)))
151150impd 414 . . . . 5 (Fun ((𝑀 Sat 𝐸)‘𝑁) → ((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))) → 𝑦 = 𝑧))
152151alrimivv 1930 . . . 4 (Fun ((𝑀 Sat 𝐸)‘𝑁) → ∀𝑦𝑧((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))) → 𝑦 = 𝑧))
153 eqeq1 2828 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
154153anbi2d 631 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
155154rexbidv 3289 . . . . . . 7 (𝑦 = 𝑧 → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
156 eqeq1 2828 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ↔ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
157156anbi2d 631 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
158157rexbidv 3289 . . . . . . 7 (𝑦 = 𝑧 → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
159155, 158orbi12d 916 . . . . . 6 (𝑦 = 𝑧 → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
160159rexbidv 3289 . . . . 5 (𝑦 = 𝑧 → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
161160mo4 2651 . . . 4 (∃*𝑦𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∀𝑦𝑧((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑧 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))) → 𝑦 = 𝑧))
162152, 161sylibr 237 . . 3 (Fun ((𝑀 Sat 𝐸)‘𝑁) → ∃*𝑦𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
163162alrimiv 1929 . 2 (Fun ((𝑀 Sat 𝐸)‘𝑁) → ∀𝑥∃*𝑦𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
164 funopab 6378 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} ↔ ∀𝑥∃*𝑦𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
165163, 164sylibr 237 1 (Fun ((𝑀 Sat 𝐸)‘𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wcel 2115  ∃*wmo 2622  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  cdif 3916  cun 3917  cin 3918  {csn 4550  cop 4556  {copab 5114  cres 5544  Fun wfun 6337  cfv 6343  (class class class)co 7149  ωcom 7574  1st c1st 7682  2nd c2nd 7683  1oc1o 8091  2oc2o 8092  m cmap 8402  𝑔cgna 32641  𝑔cgol 32642   Sat csat 32643
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-om 7575  df-1st 7684  df-2nd 7685  df-1o 8098  df-2o 8099  df-gona 32648  df-goal 32649
This theorem is referenced by:  satffunlem1  32714
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