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Theorem sat1el2xp 32861
 Description: The first component of an element of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation is a member of a doubled Cartesian product. (Contributed by AV, 17-Sep-2023.)
Assertion
Ref Expression
sat1el2xp (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
Distinct variable groups:   𝑤,𝑁   𝑎,𝑏,𝑤
Allowed substitution hints:   𝑁(𝑎,𝑏)

Proof of Theorem sat1el2xp
Dummy variables 𝑥 𝑓 𝑖 𝑗 𝑢 𝑣 𝑟 𝑠 𝑡 𝑦 𝑒 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6662 . . 3 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
21raleqdv 3329 . 2 (𝑥 = ∅ → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3 fveq2 6662 . . 3 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
43raleqdv 3329 . 2 (𝑥 = 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
5 fveq2 6662 . . 3 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
65raleqdv 3329 . 2 (𝑥 = suc 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
7 fveq2 6662 . . 3 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
87raleqdv 3329 . 2 (𝑥 = 𝑁 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
9 eqeq1 2762 . . . . . . . 8 (𝑥 = (1st𝑤) → (𝑥 = (𝑖𝑔𝑗) ↔ (1st𝑤) = (𝑖𝑔𝑗)))
1092rexbidv 3224 . . . . . . 7 (𝑥 = (1st𝑤) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
1110anbi2d 631 . . . . . 6 (𝑥 = (1st𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
12 eqeq1 2762 . . . . . . 7 (𝑧 = (2nd𝑤) → (𝑧 = ∅ ↔ (2nd𝑤) = ∅))
1312anbi1d 632 . . . . . 6 (𝑧 = (2nd𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)) ↔ ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
1411, 13elopabi 7769 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} → ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
15 goel 32829 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1615eqeq2d 2769 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) ↔ (1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
17 omex 9144 . . . . . . . . . . 11 ω ∈ V
1817, 17pm3.2i 474 . . . . . . . . . 10 (ω ∈ V ∧ ω ∈ V)
19 peano1 7605 . . . . . . . . . . . 12 ∅ ∈ ω
2019a1i 11 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ ω)
21 opelxpi 5564 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
2220, 21opelxpd 5565 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω)))
23 xpeq12 5552 . . . . . . . . . . . . 13 ((𝑎 = ω ∧ 𝑏 = ω) → (𝑎 × 𝑏) = (ω × ω))
2423xpeq2d 5557 . . . . . . . . . . . 12 ((𝑎 = ω ∧ 𝑏 = ω) → (ω × (𝑎 × 𝑏)) = (ω × (ω × ω)))
2524eleq2d 2837 . . . . . . . . . . 11 ((𝑎 = ω ∧ 𝑏 = ω) → (⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω))))
2625spc2egv 3520 . . . . . . . . . 10 ((ω ∈ V ∧ ω ∈ V) → (⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω)) → ∃𝑎𝑏⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
2718, 22, 26mpsyl 68 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∃𝑎𝑏⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏)))
28 eleq1 2839 . . . . . . . . . 10 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
29282exbidv 1925 . . . . . . . . 9 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
3027, 29syl5ibrcom 250 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3116, 30sylbid 243 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3231rexlimivv 3216 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3332adantl 485 . . . . 5 (((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3414, 33syl 17 . . . 4 (𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
35 satf00 32856 . . . 4 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
3634, 35eleq2s 2870 . . 3 (𝑤 ∈ ((∅ Sat ∅)‘∅) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3736rgen 3080 . 2 𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))
38 omsucelsucb 8109 . . . . . . . . . . 11 (𝑦 ∈ ω ↔ suc 𝑦 ∈ suc ω)
39 satf0sucom 32855 . . . . . . . . . . 11 (suc 𝑦 ∈ suc ω → ((∅ Sat ∅)‘suc 𝑦) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦))
4038, 39sylbi 220 . . . . . . . . . 10 (𝑦 ∈ ω → ((∅ Sat ∅)‘suc 𝑦) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦))
4140adantr 484 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((∅ Sat ∅)‘suc 𝑦) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦))
42 nnon 7590 . . . . . . . . . . . 12 (𝑦 ∈ ω → 𝑦 ∈ On)
43 rdgsuc 8075 . . . . . . . . . . . 12 (𝑦 ∈ On → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦) = ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘(rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦)))
4442, 43syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦) = ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘(rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦)))
4544adantr 484 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦) = ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘(rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦)))
46 elelsuc 6245 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ suc ω)
47 satf0sucom 32855 . . . . . . . . . . . . . 14 (𝑦 ∈ suc ω → ((∅ Sat ∅)‘𝑦) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦))
4846, 47syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ ω → ((∅ Sat ∅)‘𝑦) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦))
4948eqcomd 2764 . . . . . . . . . . . 12 (𝑦 ∈ ω → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦) = ((∅ Sat ∅)‘𝑦))
5049fveq2d 6666 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘(rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦)) = ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘((∅ Sat ∅)‘𝑦)))
5150adantr 484 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘(rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑦)) = ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘((∅ Sat ∅)‘𝑦)))
52 eqidd 2759 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
53 id 22 . . . . . . . . . . . . 13 (𝑓 = ((∅ Sat ∅)‘𝑦) → 𝑓 = ((∅ Sat ∅)‘𝑦))
54 rexeq 3324 . . . . . . . . . . . . . . . . 17 (𝑓 = ((∅ Sat ∅)‘𝑦) → (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
5554orbi1d 914 . . . . . . . . . . . . . . . 16 (𝑓 = ((∅ Sat ∅)‘𝑦) → ((∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
5655rexeqbi1dv 3322 . . . . . . . . . . . . . . 15 (𝑓 = ((∅ Sat ∅)‘𝑦) → (∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
5756anbi2d 631 . . . . . . . . . . . . . 14 (𝑓 = ((∅ Sat ∅)‘𝑦) → ((𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ↔ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
5857opabbidv 5101 . . . . . . . . . . . . 13 (𝑓 = ((∅ Sat ∅)‘𝑦) → {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} = {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})
5953, 58uneq12d 4071 . . . . . . . . . . . 12 (𝑓 = ((∅ Sat ∅)‘𝑦) → (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
6059adantl 485 . . . . . . . . . . 11 (((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) ∧ 𝑓 = ((∅ Sat ∅)‘𝑦)) → (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
61 fvexd 6677 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((∅ Sat ∅)‘𝑦) ∈ V)
6217a1i 11 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ω ∈ V)
63 satf0suclem 32857 . . . . . . . . . . . . 13 ((((∅ Sat ∅)‘𝑦) ∈ V ∧ ((∅ Sat ∅)‘𝑦) ∈ V ∧ ω ∈ V) → {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ∈ V)
6461, 61, 62, 63syl3anc 1368 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ∈ V)
65 unexg 7475 . . . . . . . . . . . 12 ((((∅ Sat ∅)‘𝑦) ∈ V ∧ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} ∈ V) → (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ∈ V)
6661, 64, 65syl2anc 587 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ∈ V)
6752, 60, 61, 66fvmptd 6770 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))‘((∅ Sat ∅)‘𝑦)) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
6845, 51, 673eqtrd 2797 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
6941, 68eqtrd 2793 . . . . . . . 8 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((∅ Sat ∅)‘suc 𝑦) = (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
7069eleq2d 2837 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ 𝑡 ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
71 elun 4056 . . . . . . 7 (𝑡 ∈ (((∅ Sat ∅)‘𝑦) ∪ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) ↔ (𝑡 ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝑡 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
7270, 71bitrdi 290 . . . . . 6 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) ↔ (𝑡 ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝑡 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})))
73 fveq2 6662 . . . . . . . . . . 11 (𝑤 = 𝑡 → (1st𝑤) = (1st𝑡))
7473eleq1d 2836 . . . . . . . . . 10 (𝑤 = 𝑡 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
75742exbidv 1925 . . . . . . . . 9 (𝑤 = 𝑡 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7675rspccv 3540 . . . . . . . 8 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7776adantl 485 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
78 fveq2 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 → (1st𝑤) = (1st𝑣))
7978eleq1d 2836 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
80792exbidv 1925 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑣 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
8180rspcva 3541 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏)))
82 sels 5305 . . . . . . . . . . . . . . . . . 18 ((1st𝑣) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑣) ∈ 𝑠)
8382exlimivv 1933 . . . . . . . . . . . . . . . . 17 (∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑣) ∈ 𝑠)
8481, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑠(1st𝑣) ∈ 𝑠)
8584expcom 417 . . . . . . . . . . . . . . 15 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑠(1st𝑣) ∈ 𝑠))
86 fveq2 6662 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (1st𝑤) = (1st𝑢))
8786eleq1d 2836 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
88872exbidv 1925 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
8988rspcva 3541 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏)))
90 sels 5305 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑢) ∈ 𝑠)
9190exlimivv 1933 . . . . . . . . . . . . . . . . . . 19 (∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑢) ∈ 𝑠)
9289, 91syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑠(1st𝑢) ∈ 𝑠)
93 eleq2w 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑟 → ((1st𝑢) ∈ 𝑠 ↔ (1st𝑢) ∈ 𝑟))
9493cbvexvw 2044 . . . . . . . . . . . . . . . . . . 19 (∃𝑠(1st𝑢) ∈ 𝑠 ↔ ∃𝑟(1st𝑢) ∈ 𝑟)
95 vex 3413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑟 ∈ V
96 vex 3413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑠 ∈ V
9795, 96pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑟 ∈ V ∧ 𝑠 ∈ V)
98 df-ov 7158 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st𝑢)⊼𝑔(1st𝑣)) = (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩)
99 df-gona 32823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑔 = (𝑒 ∈ (V × V) ↦ ⟨1o, 𝑒⟩)
100 opeq2 4766 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = ⟨(1st𝑢), (1st𝑣)⟩ → ⟨1o, 𝑒⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
101 opelvvg 5567 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (V × V))
102 opex 5327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V)
10499, 100, 101, 103fvmptd3 6786 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
10598, 104syl5eq 2805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
106 1onn 8280 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1o ∈ ω
107106a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → 1o ∈ ω)
108 opelxpi 5564 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (𝑟 × 𝑠))
109107, 108opelxpd 5565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ (ω × (𝑟 × 𝑠)))
110105, 109eqeltrd 2852 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠)))
111 xpeq12 5552 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 = 𝑟𝑏 = 𝑠) → (𝑎 × 𝑏) = (𝑟 × 𝑠))
112111xpeq2d 5557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 = 𝑟𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (𝑟 × 𝑠)))
113112eleq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 = 𝑟𝑏 = 𝑠) → (((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠))))
114113spc2egv 3520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑟 ∈ V ∧ 𝑠 ∈ V) → (((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠)) → ∃𝑎𝑏((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
11597, 110, 114mpsyl 68 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ∃𝑎𝑏((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏)))
116 eleq1 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
1171162exbidv 1925 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
118115, 117syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
119118ex 416 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑢) ∈ 𝑟 → ((1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
120119exlimdv 1934 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑢) ∈ 𝑟 → (∃𝑠(1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
121120com23 86 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) ∈ 𝑟 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
122121exlimiv 1931 . . . . . . . . . . . . . . . . . . 19 (∃𝑟(1st𝑢) ∈ 𝑟 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
12394, 122sylbi 220 . . . . . . . . . . . . . . . . . 18 (∃𝑠(1st𝑢) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
12492, 123syl 17 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
125124expcom 417 . . . . . . . . . . . . . . . 16 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
126125com24 95 . . . . . . . . . . . . . . 15 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
12785, 126syld 47 . . . . . . . . . . . . . 14 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
128127adantl 485 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
129128com14 96 . . . . . . . . . . . 12 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
130129rexlimdv 3207 . . . . . . . . . . 11 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
13117, 96pm3.2i 474 . . . . . . . . . . . . . . . . . . . . 21 (ω ∈ V ∧ 𝑠 ∈ V)
132 df-goal 32824 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑔𝑖(1st𝑢) = ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩
133 2onn 8281 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2o ∈ ω
134133a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → 2o ∈ ω)
135 opelxpi 5564 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ ω ∧ (1st𝑢) ∈ 𝑠) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
136135ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
137134, 136opelxpd 5565 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ ∈ (ω × (ω × 𝑠)))
138132, 137eqeltrid 2856 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠)))
1391383adant3 1129 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠)))
140 eleq1 2839 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → ((1st𝑡) ∈ (ω × (ω × 𝑠)) ↔ ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠))))
1411403ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ((1st𝑡) ∈ (ω × (ω × 𝑠)) ↔ ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠))))
142139, 141mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → (1st𝑡) ∈ (ω × (ω × 𝑠)))
143 xpeq12 5552 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (𝑎 × 𝑏) = (ω × 𝑠))
144143xpeq2d 5557 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (ω × 𝑠)))
145144eleq2d 2837 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = ω ∧ 𝑏 = 𝑠) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (ω × 𝑠))))
146145spc2egv 3520 . . . . . . . . . . . . . . . . . . . . 21 ((ω ∈ V ∧ 𝑠 ∈ V) → ((1st𝑡) ∈ (ω × (ω × 𝑠)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
147131, 142, 146mpsyl 68 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))
1481473exp 1116 . . . . . . . . . . . . . . . . . . 19 ((1st𝑢) ∈ 𝑠 → (𝑖 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
149148com23 86 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) ∈ 𝑠 → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
150149a1d 25 . . . . . . . . . . . . . . . . 17 ((1st𝑢) ∈ 𝑠 → (𝑦 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
151150exlimiv 1931 . . . . . . . . . . . . . . . 16 (∃𝑠(1st𝑢) ∈ 𝑠 → (𝑦 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
15292, 151syl 17 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑦 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
153152ex 416 . . . . . . . . . . . . . 14 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑦 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))))
154153impcomd 415 . . . . . . . . . . . . 13 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → (𝑖 ∈ ω → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
155154com24 95 . . . . . . . . . . . 12 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → (𝑖 ∈ ω → ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))))
156155rexlimdv 3207 . . . . . . . . . . 11 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → (∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
157130, 156jaod 856 . . . . . . . . . 10 (𝑢 ∈ ((∅ Sat ∅)‘𝑦) → ((∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
158157rexlimiv 3204 . . . . . . . . 9 (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
159158adantl 485 . . . . . . . 8 (((2nd𝑡) = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢))) → ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
160 eqeq1 2762 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ (1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
161160rexbidv 3221 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
162 eqeq1 2762 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
163162rexbidv 3221 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
164161, 163orbi12d 916 . . . . . . . . . . 11 (𝑥 = (1st𝑡) → ((∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢))))
165164rexbidv 3221 . . . . . . . . . 10 (𝑥 = (1st𝑡) → (∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢))))
166165anbi2d 631 . . . . . . . . 9 (𝑥 = (1st𝑡) → ((𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ↔ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)))))
167 eqeq1 2762 . . . . . . . . . 10 (𝑧 = (2nd𝑡) → (𝑧 = ∅ ↔ (2nd𝑡) = ∅))
168167anbi1d 632 . . . . . . . . 9 (𝑧 = (2nd𝑡) → ((𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢))) ↔ ((2nd𝑡) = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)))))
169166, 168elopabi 7769 . . . . . . . 8 (𝑡 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} → ((2nd𝑡) = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢))))
170159, 169syl11 33 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))} → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
17177, 170jaod 856 . . . . . 6 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ((𝑡 ∈ ((∅ Sat ∅)‘𝑦) ∨ 𝑡 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑦)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
17272, 171sylbid 243 . . . . 5 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
173172ex 416 . . . 4 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
174173ralrimdv 3117 . . 3 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
17575cbvralvw 3361 . . 3 (∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))
176174, 175syl6ibr 255 . 2 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → ∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
1772, 4, 6, 8, 37, 176finds 7613 1 (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∪ cun 3858  ∅c0 4227  ⟨cop 4531  {copab 5097   ↦ cmpt 5115   × cxp 5525  Oncon0 6173  suc csuc 6175  ‘cfv 6339  (class class class)co 7155  ωcom 7584  1st c1st 7696  2nd c2nd 7697  reccrdg 8060  1oc1o 8110  2oc2o 8111  ∈𝑔cgoe 32815  ⊼𝑔cgna 32816  ∀𝑔cgol 32817   Sat csat 32818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-map 8423  df-goel 32822  df-gona 32823  df-goal 32824  df-sat 32825 This theorem is referenced by: (None)
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