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Theorem satfdm 32620
 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 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ∀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.
StepHypRef Expression
1 fveq2 6673 . . . . . . 7 (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘∅))
21dmeqd 5777 . . . . . 6 (𝑥 = ∅ → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘∅))
3 fveq2 6673 . . . . . . 7 (𝑥 = ∅ → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘∅))
43dmeqd 5777 . . . . . 6 (𝑥 = ∅ → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘∅))
52, 4eqeq12d 2840 . . . . 5 (𝑥 = ∅ → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅)))
65imbi2d 343 . . . 4 (𝑥 = ∅ → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅))))
7 fveq2 6673 . . . . . . 7 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
87dmeqd 5777 . . . . . 6 (𝑥 = 𝑦 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘𝑦))
9 fveq2 6673 . . . . . . 7 (𝑥 = 𝑦 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘𝑦))
109dmeqd 5777 . . . . . 6 (𝑥 = 𝑦 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑦))
118, 10eqeq12d 2840 . . . . 5 (𝑥 = 𝑦 → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
1211imbi2d 343 . . . 4 (𝑥 = 𝑦 → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦))))
13 fveq2 6673 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
1413dmeqd 5777 . . . . . 6 (𝑥 = suc 𝑦 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘suc 𝑦))
15 fveq2 6673 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘suc 𝑦))
1615dmeqd 5777 . . . . . 6 (𝑥 = suc 𝑦 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))
1714, 16eqeq12d 2840 . . . . 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)))
1817imbi2d 343 . . . 4 (𝑥 = suc 𝑦 → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
19 fveq2 6673 . . . . . . 7 (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑛))
2019dmeqd 5777 . . . . . 6 (𝑥 = 𝑛 → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑀 Sat 𝐸)‘𝑛))
21 fveq2 6673 . . . . . . 7 (𝑥 = 𝑛 → ((𝑁 Sat 𝐹)‘𝑥) = ((𝑁 Sat 𝐹)‘𝑛))
2221dmeqd 5777 . . . . . 6 (𝑥 = 𝑛 → dom ((𝑁 Sat 𝐹)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑛))
2320, 22eqeq12d 2840 . . . . 5 (𝑥 = 𝑛 → (dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥) ↔ dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛)))
2423imbi2d 343 . . . 4 (𝑥 = 𝑛 → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑥) = dom ((𝑁 Sat 𝐹)‘𝑥)) ↔ (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))))
25 rexcom4 3252 . . . . . . . . . 10 (∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑦𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}))
2625rexbii 3250 . . . . . . . . 9 (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑦𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}))
27 ovex 7192 . . . . . . . . . . . . . . . 16 (𝑀m ω) ∈ V
2827rabex 5238 . . . . . . . . . . . . . . 15 {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)} ∈ V
2928isseti 3511 . . . . . . . . . . . . . 14 𝑦 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}
30 ovex 7192 . . . . . . . . . . . . . . . 16 (𝑁m ω) ∈ V
3130rabex 5238 . . . . . . . . . . . . . . 15 {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)} ∈ V
3231isseti 3511 . . . . . . . . . . . . . 14 𝑧 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}
3329, 322th 266 . . . . . . . . . . . . 13 (∃𝑦 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)} ↔ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})
3433anbi2i 624 . . . . . . . . . . . 12 ((𝑥 = (𝑢𝑔𝑣) ∧ ∃𝑦 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ (𝑥 = (𝑢𝑔𝑣) ∧ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
35 19.42v 1953 . . . . . . . . . . . 12 (∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ (𝑥 = (𝑢𝑔𝑣) ∧ ∃𝑦 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}))
36 19.42v 1953 . . . . . . . . . . . 12 (∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}) ↔ (𝑥 = (𝑢𝑔𝑣) ∧ ∃𝑧 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
3734, 35, 363bitr4i 305 . . . . . . . . . . 11 (∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
3837rexbii 3250 . . . . . . . . . 10 (∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
3938rexbii 3250 . . . . . . . . 9 (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑦(𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
40 rexcom4 3252 . . . . . . . . 9 (∃𝑢 ∈ ω ∃𝑦𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}))
4126, 39, 403bitr3ri 304 . . . . . . . 8 (∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
42 rexcom4 3252 . . . . . . . . 9 (∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}) ↔ ∃𝑧𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
4342rexbii 3250 . . . . . . . 8 (∃𝑢 ∈ ω ∃𝑣 ∈ ω ∃𝑧(𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑧𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
4441, 43bitri 277 . . . . . . 7 (∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑢 ∈ ω ∃𝑧𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
45 rexcom4 3252 . . . . . . 7 (∃𝑢 ∈ ω ∃𝑧𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}) ↔ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
4644, 45bitri 277 . . . . . 6 (∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)}) ↔ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)}))
4746abbii 2889 . . . . 5 {𝑥 ∣ ∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})} = {𝑥 ∣ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})}
48 eqid 2824 . . . . . . . . 9 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4948satfv0 32609 . . . . . . . 8 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})})
5049dmeqd 5777 . . . . . . 7 ((𝑀𝑉𝐸𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})})
51 dmopab 5787 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})} = {𝑥 ∣ ∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})}
5250, 51syl6eq 2875 . . . . . 6 ((𝑀𝑉𝐸𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = {𝑥 ∣ ∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})})
5352adantr 483 . . . . 5 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = {𝑥 ∣ ∃𝑦𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑢)𝐸(𝑎𝑣)})})
54 eqid 2824 . . . . . . . . 9 (𝑁 Sat 𝐹) = (𝑁 Sat 𝐹)
5554satfv0 32609 . . . . . . . 8 ((𝑁𝑋𝐹𝑌) → ((𝑁 Sat 𝐹)‘∅) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})})
5655dmeqd 5777 . . . . . . 7 ((𝑁𝑋𝐹𝑌) → dom ((𝑁 Sat 𝐹)‘∅) = dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})})
57 dmopab 5787 . . . . . . 7 dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})} = {𝑥 ∣ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})}
5856, 57syl6eq 2875 . . . . . 6 ((𝑁𝑋𝐹𝑌) → dom ((𝑁 Sat 𝐹)‘∅) = {𝑥 ∣ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})})
5958adantl 484 . . . . 5 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑁 Sat 𝐹)‘∅) = {𝑥 ∣ ∃𝑧𝑢 ∈ ω ∃𝑣 ∈ ω (𝑥 = (𝑢𝑔𝑣) ∧ 𝑧 = {𝑎 ∈ (𝑁m ω) ∣ (𝑎𝑢)𝐹(𝑎𝑣)})})
6047, 53, 593eqtr4a 2885 . . . 4 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘∅) = dom ((𝑁 Sat 𝐹)‘∅))
61 pm2.27 42 . . . . . . . 8 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
6261adantl 484 . . . . . . 7 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)))
63 simpr 487 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦))
64 simprl 769 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (𝑀𝑉𝐸𝑊))
65 simpl 485 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → 𝑦 ∈ ω)
66 df-3an 1085 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊𝑦 ∈ ω) ↔ ((𝑀𝑉𝐸𝑊) ∧ 𝑦 ∈ ω))
6764, 65, 66sylanbrc 585 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (𝑀𝑉𝐸𝑊𝑦 ∈ ω))
68 satfdmlem 32619 . . . . . . . . . . . . . . . . 17 (((𝑀𝑉𝐸𝑊𝑦 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
6967, 68sylan 582 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
70 simprr 771 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (𝑁𝑋𝐹𝑌))
71 df-3an 1085 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝐹𝑌𝑦 ∈ ω) ↔ ((𝑁𝑋𝐹𝑌) ∧ 𝑦 ∈ ω))
7270, 65, 71sylanbrc 585 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (𝑁𝑋𝐹𝑌𝑦 ∈ 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))
7473eqcomd 2830 . . . . . . . . . . . . . . . . 17 (dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦) → dom ((𝑁 Sat 𝐹)‘𝑦) = dom ((𝑀 Sat 𝐸)‘𝑦))
75 satfdmlem 32619 . . . . . . . . . . . . . . . . 17 (((𝑁𝑋𝐹𝑌𝑦 ∈ ω) ∧ dom ((𝑁 Sat 𝐹)‘𝑦) = dom ((𝑀 Sat 𝐸)‘𝑦)) → (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
7672, 74, 75syl2an 597 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
7769, 76impbid 214 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
7827difexi 5235 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V
7978isseti 3511 . . . . . . . . . . . . . . . . . . . 20 𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))
8079biantru 532 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
8180bicomi 226 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
8281rexbii 3250 . . . . . . . . . . . . . . . . 17 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
8327rabex 5238 . . . . . . . . . . . . . . . . . . . . 21 {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V
8483isseti 3511 . . . . . . . . . . . . . . . . . . . 20 𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}
8584biantru 532 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
8685bicomi 226 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ 𝑥 = ∀𝑔𝑖(1st𝑢))
8786rexbii 3250 . . . . . . . . . . . . . . . . 17 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))
8882, 87orbi12i 911 . . . . . . . . . . . . . . . 16 ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
8988rexbii 3250 . . . . . . . . . . . . . . 15 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
9030difexi 5235 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏))) ∈ V
9190isseti 3511 . . . . . . . . . . . . . . . . . . . 20 𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))
9291biantru 532 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ↔ (𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
9392bicomi 226 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ 𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))
9493rexbii 3250 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))
9530rabex 5238 . . . . . . . . . . . . . . . . . . . . 21 {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)} ∈ V
9695isseti 3511 . . . . . . . . . . . . . . . . . . . 20 𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}
9796biantru 532 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ∀𝑔𝑖(1st𝑎) ↔ (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
9897bicomi 226 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ 𝑥 = ∀𝑔𝑖(1st𝑎))
9998rexbii 3250 . . . . . . . . . . . . . . . . 17 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))
10094, 99orbi12i 911 . . . . . . . . . . . . . . . 16 ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))
101100rexbii 3250 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))
10277, 89, 1013bitr4g 316 . . . . . . . . . . . . . 14 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))))
103 19.42v 1953 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑤(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
104103bicomi 226 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑤(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
105104rexbii 3250 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
106 rexcom4 3252 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑤𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
107105, 106bitri 277 . . . . . . . . . . . . . . . . . 18 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑤𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
108 19.42v 1953 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑤(𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
109108bicomi 226 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑤(𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
110109rexbii 3250 . . . . . . . . . . . . . . . . . . 19 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω ∃𝑤(𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
111 rexcom4 3252 . . . . . . . . . . . . . . . . . . 19 (∃𝑖 ∈ ω ∃𝑤(𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑤𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
112110, 111bitri 277 . . . . . . . . . . . . . . . . . 18 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑤𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
113107, 112orbi12i 911 . . . . . . . . . . . . . . . . 17 ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑤𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑤𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
114 19.43 1882 . . . . . . . . . . . . . . . . . 18 (∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑤𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑤𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
115114bicomi 226 . . . . . . . . . . . . . . . . 17 ((∃𝑤𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑤𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
116113, 115bitri 277 . . . . . . . . . . . . . . . 16 ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
117116rexbii 3250 . . . . . . . . . . . . . . 15 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
118 rexcom4 3252 . . . . . . . . . . . . . . 15 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)∃𝑤(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑤𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
119117, 118bitri 277 . . . . . . . . . . . . . 14 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ ∃𝑤 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ ∃𝑤 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑤𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
120 19.42v 1953 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑧(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ (𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
121120bicomi 226 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ ∃𝑧(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
122121rexbii 3250 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
123 rexcom4 3252 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ ∃𝑧𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
124122, 123bitri 277 . . . . . . . . . . . . . . . . . 18 (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ↔ ∃𝑧𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))))
125 19.42v 1953 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑧(𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
126125bicomi 226 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ ∃𝑧(𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
127126rexbii 3250 . . . . . . . . . . . . . . . . . . 19 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ ∃𝑖 ∈ ω ∃𝑧(𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
128 rexcom4 3252 . . . . . . . . . . . . . . . . . . 19 (∃𝑖 ∈ ω ∃𝑧(𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ ∃𝑧𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
129127, 128bitri 277 . . . . . . . . . . . . . . . . . 18 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}) ↔ ∃𝑧𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))
130124, 129orbi12i 911 . . . . . . . . . . . . . . . . 17 ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ (∃𝑧𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑧𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
131 19.43 1882 . . . . . . . . . . . . . . . . . 18 (∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ (∃𝑧𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑧𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
132131bicomi 226 . . . . . . . . . . . . . . . . 17 ((∃𝑧𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑧𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
133130, 132bitri 277 . . . . . . . . . . . . . . . 16 ((∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
134133rexbii 3250 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
135 rexcom4 3252 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)∃𝑧(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑧𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
136134, 135bitri 277 . . . . . . . . . . . . . 14 (∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ ∃𝑧 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ ∃𝑧 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})) ↔ ∃𝑧𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)})))
137102, 119, 1363bitr3g 315 . . . . . . . . . . . . 13 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (∃𝑤𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑧𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))))
138137abbidv 2888 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → {𝑥 ∣ ∃𝑤𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {𝑥 ∣ ∃𝑧𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))})
139 dmopab 5787 . . . . . . . . . . . 12 dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {𝑥 ∣ ∃𝑤𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}
140 dmopab 5787 . . . . . . . . . . . 12 dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))} = {𝑥 ∣ ∃𝑧𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}
141138, 139, 1403eqtr4g 2884 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))})
14263, 141uneq12d 4143 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = (dom ((𝑁 Sat 𝐹)‘𝑦) ∪ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}))
143 dmun 5782 . . . . . . . . . 10 dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = (dom ((𝑀 Sat 𝐸)‘𝑦) ∪ dom {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
144 dmun 5782 . . . . . . . . . 10 dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}) = (dom ((𝑁 Sat 𝐹)‘𝑦) ∪ dom {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))})
145142, 143, 1443eqtr4g 2884 . . . . . . . . 9 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}))
146 simpl 485 . . . . . . . . . . . . . 14 ((𝑀𝑉𝐸𝑊) → 𝑀𝑉)
147146adantr 483 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → 𝑀𝑉)
148 simpr 487 . . . . . . . . . . . . . 14 ((𝑀𝑉𝐸𝑊) → 𝐸𝑊)
149148adantr 483 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → 𝐸𝑊)
15048satfvsuc 32612 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊𝑦 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑦) = (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
151147, 149, 65, 150syl2an23an 1419 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → ((𝑀 Sat 𝐸)‘suc 𝑦) = (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
152151dmeqd 5777 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom (((𝑀 Sat 𝐸)‘𝑦) ∪ {⟨𝑥, 𝑤⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑦)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑦)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
153 simprl 769 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → 𝑁𝑋)
154 simprr 771 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → 𝐹𝑌)
15554satfvsuc 32612 . . . . . . . . . . . . 13 ((𝑁𝑋𝐹𝑌𝑦 ∈ ω) → ((𝑁 Sat 𝐹)‘suc 𝑦) = (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}))
156153, 154, 65, 155syl2an23an 1419 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → ((𝑁 Sat 𝐹)‘suc 𝑦) = (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}))
157156dmeqd 5777 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → dom ((𝑁 Sat 𝐹)‘suc 𝑦) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))}))
158152, 157eqeq12d 2840 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))})))
159158adantr 483 . . . . . . . . 9 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑤 = {𝑚 ∈ (𝑀m ω) ∣ ∀𝑓𝑀 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = dom (((𝑁 Sat 𝐹)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑦)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑦)(𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∧ 𝑧 = ((𝑁m ω) ∖ ((2nd𝑎) ∩ (2nd𝑏)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑎) ∧ 𝑧 = {𝑚 ∈ (𝑁m ω) ∣ ∀𝑓𝑁 ({⟨𝑖, 𝑓⟩} ∪ (𝑚 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑎)}))})))
160145, 159mpbird 259 . . . . . . . 8 (((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) ∧ dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))
161160ex 415 . . . . . . 7 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → (dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦)))
16262, 161syld 47 . . . . . 6 ((𝑦 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌))) → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦)))
163162ex 415 . . . . 5 (𝑦 ∈ ω → (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
164163com23 86 . . . 4 (𝑦 ∈ ω → ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑦) = dom ((𝑁 Sat 𝐹)‘𝑦)) → (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘suc 𝑦) = dom ((𝑁 Sat 𝐹)‘suc 𝑦))))
1656, 12, 18, 24, 60, 164finds 7611 . . 3 (𝑛 ∈ ω → (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛)))
166165impcom 410 . 2 ((((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) ∧ 𝑛 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
167166ralrimiva 3185 1 (((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142  {crab 3145   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938  ∅c0 4294  {csn 4570  ⟨cop 4576   class class class wbr 5069  {copab 5131  dom cdm 5558   ↾ cres 5560  suc csuc 6196  ‘cfv 6358  (class class class)co 7159  ωcom 7583  1st c1st 7690  2nd c2nd 7691   ↑m cmap 8409  ∈𝑔cgoe 32584  ⊼𝑔cgna 32585  ∀𝑔cgol 32586   Sat csat 32587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-goel 32591  df-goal 32593  df-sat 32594 This theorem is referenced by:  satfdmfmla  32651
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