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Theorem sat1el2xp 33973
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 6842 . . 3 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
21raleqdv 3313 . 2 (𝑥 = ∅ → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3 fveq2 6842 . . 3 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
43raleqdv 3313 . 2 (𝑥 = 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
5 fveq2 6842 . . 3 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
65raleqdv 3313 . 2 (𝑥 = suc 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
7 fveq2 6842 . . 3 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
87raleqdv 3313 . 2 (𝑥 = 𝑁 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
9 eqeq1 2740 . . . . . . . 8 (𝑥 = (1st𝑤) → (𝑥 = (𝑖𝑔𝑗) ↔ (1st𝑤) = (𝑖𝑔𝑗)))
1092rexbidv 3213 . . . . . . 7 (𝑥 = (1st𝑤) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
1110anbi2d 629 . . . . . 6 (𝑥 = (1st𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
12 eqeq1 2740 . . . . . . 7 (𝑧 = (2nd𝑤) → (𝑧 = ∅ ↔ (2nd𝑤) = ∅))
1312anbi1d 630 . . . . . 6 (𝑧 = (2nd𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)) ↔ ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
1411, 13elopabi 7994 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} → ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
15 goel 33941 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1615eqeq2d 2747 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) ↔ (1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
17 omex 9579 . . . . . . . . . . 11 ω ∈ V
1817, 17pm3.2i 471 . . . . . . . . . 10 (ω ∈ V ∧ ω ∈ V)
19 peano1 7825 . . . . . . . . . . . 12 ∅ ∈ ω
2019a1i 11 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ ω)
21 opelxpi 5670 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
2220, 21opelxpd 5671 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω)))
23 xpeq12 5658 . . . . . . . . . . . . 13 ((𝑎 = ω ∧ 𝑏 = ω) → (𝑎 × 𝑏) = (ω × ω))
2423xpeq2d 5663 . . . . . . . . . . . 12 ((𝑎 = ω ∧ 𝑏 = ω) → (ω × (𝑎 × 𝑏)) = (ω × (ω × ω)))
2524eleq2d 2823 . . . . . . . . . . 11 ((𝑎 = ω ∧ 𝑏 = ω) → (⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω))))
2625spc2egv 3558 . . . . . . . . . 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 2825 . . . . . . . . . 10 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
29282exbidv 1927 . . . . . . . . 9 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
3027, 29syl5ibrcom 246 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3116, 30sylbid 239 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3231rexlimivv 3196 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3332adantl 482 . . . . 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 33968 . . . 4 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
3634, 35eleq2s 2856 . . 3 (𝑤 ∈ ((∅ Sat ∅)‘∅) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3736rgen 3066 . 2 𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))
38 omsucelsucb 8404 . . . . . . . . . . 11 (𝑦 ∈ ω ↔ suc 𝑦 ∈ suc ω)
39 satf0sucom 33967 . . . . . . . . . . 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 216 . . . . . . . . . 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 481 . . . . . . . . 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 7808 . . . . . . . . . . . 12 (𝑦 ∈ ω → 𝑦 ∈ On)
43 rdgsuc 8370 . . . . . . . . . . . 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 481 . . . . . . . . . 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 6390 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ suc ω)
47 satf0sucom 33967 . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . 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 6846 . . . . . . . . . . 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 481 . . . . . . . . . 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 2737 . . . . . . . . . . 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 3310 . . . . . . . . . . . . . . . . 17 (𝑓 = ((∅ Sat ∅)‘𝑦) → (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
5554orbi1d 915 . . . . . . . . . . . . . . . 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 3308 . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . 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 5171 . . . . . . . . . . . . 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 4124 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 6857 . . . . . . . . . . 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 33969 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 7683 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 6955 . . . . . . . . . 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 2780 . . . . . . . . 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 2776 . . . . . . . 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 2823 . . . . . . 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 4108 . . . . . . 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 286 . . . . . 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 6842 . . . . . . . . . . 11 (𝑤 = 𝑡 → (1st𝑤) = (1st𝑡))
7473eleq1d 2822 . . . . . . . . . 10 (𝑤 = 𝑡 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
75742exbidv 1927 . . . . . . . . 9 (𝑤 = 𝑡 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7675rspccv 3578 . . . . . . . 8 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7776adantl 482 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
78 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 → (1st𝑤) = (1st𝑣))
7978eleq1d 2822 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
80792exbidv 1927 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑣 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
8180rspcva 3579 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏)))
82 sels 5395 . . . . . . . . . . . . . . . . . 18 ((1st𝑣) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑣) ∈ 𝑠)
8382exlimivv 1935 . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 15 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑠(1st𝑣) ∈ 𝑠))
86 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (1st𝑤) = (1st𝑢))
8786eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
88872exbidv 1927 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
8988rspcva 3579 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏)))
90 sels 5395 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑢) ∈ 𝑠)
9190exlimivv 1935 . . . . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑟 → ((1st𝑢) ∈ 𝑠 ↔ (1st𝑢) ∈ 𝑟))
9493cbvexvw 2040 . . . . . . . . . . . . . . . . . . 19 (∃𝑠(1st𝑢) ∈ 𝑠 ↔ ∃𝑟(1st𝑢) ∈ 𝑟)
95 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑟 ∈ V
96 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑠 ∈ V
9795, 96pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑟 ∈ V ∧ 𝑠 ∈ V)
98 df-ov 7360 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st𝑢)⊼𝑔(1st𝑣)) = (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩)
99 df-gona 33935 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑔 = (𝑒 ∈ (V × V) ↦ ⟨1o, 𝑒⟩)
100 opeq2 4831 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = ⟨(1st𝑢), (1st𝑣)⟩ → ⟨1o, 𝑒⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
101 opelvvg 5673 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (V × V))
102 opex 5421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V)
10499, 100, 101, 103fvmptd3 6971 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
10598, 104eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
106 1onn 8586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1o ∈ ω
107106a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → 1o ∈ ω)
108 opelxpi 5670 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (𝑟 × 𝑠))
109107, 108opelxpd 5671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ (ω × (𝑟 × 𝑠)))
110105, 109eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠)))
111 xpeq12 5658 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 = 𝑟𝑏 = 𝑠) → (𝑎 × 𝑏) = (𝑟 × 𝑠))
112111xpeq2d 5663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 = 𝑟𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (𝑟 × 𝑠)))
113112eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 = 𝑟𝑏 = 𝑠) → (((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠))))
114113spc2egv 3558 . . . . . . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
1171162exbidv 1927 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
118115, 117syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
119118ex 413 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑢) ∈ 𝑟 → ((1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
120119exlimdv 1936 . . . . . . . . . . . . . . . . . . . . 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 1933 . . . . . . . . . . . . . . . . . . 19 (∃𝑟(1st𝑢) ∈ 𝑟 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
12394, 122sylbi 216 . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 3150 . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . 21 (ω ∈ V ∧ 𝑠 ∈ V)
132 df-goal 33936 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑔𝑖(1st𝑢) = ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩
133 2onn 8588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2o ∈ ω
134133a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → 2o ∈ ω)
135 opelxpi 5670 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ ω ∧ (1st𝑢) ∈ 𝑠) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
136135ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
137134, 136opelxpd 5671 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ ∈ (ω × (ω × 𝑠)))
138132, 137eqeltrid 2842 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠)))
1391383adant3 1132 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠)))
140 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑡) = ∀𝑔𝑖(1st𝑢) → ((1st𝑡) ∈ (ω × (ω × 𝑠)) ↔ ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠))))
1411403ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → ((1st𝑡) ∈ (ω × (ω × 𝑠)) ↔ ∀𝑔𝑖(1st𝑢) ∈ (ω × (ω × 𝑠))))
142139, 141mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → (1st𝑡) ∈ (ω × (ω × 𝑠)))
143 xpeq12 5658 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (𝑎 × 𝑏) = (ω × 𝑠))
144143xpeq2d 5663 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (ω × 𝑠)))
145144eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = ω ∧ 𝑏 = 𝑠) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (ω × 𝑠))))
146145spc2egv 3558 . . . . . . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . . . . . 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 1933 . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 3150 . . . . . . . . . . 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 857 . . . . . . . . . 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 3145 . . . . . . . . 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 482 . . . . . . . 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 2740 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ (1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
161160rexbidv 3175 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
162 eqeq1 2740 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
163162rexbidv 3175 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
164161, 163orbi12d 917 . . . . . . . . . . 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 3175 . . . . . . . . . 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 629 . . . . . . . . 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 2740 . . . . . . . . . 10 (𝑧 = (2nd𝑡) → (𝑧 = ∅ ↔ (2nd𝑡) = ∅))
168167anbi1d 630 . . . . . . . . 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 7994 . . . . . . . 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 857 . . . . . 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 239 . . . . 5 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
173172ex 413 . . . 4 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
174173ralrimdv 3149 . . 3 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
17575cbvralvw 3225 . . 3 (∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))
176174, 175syl6ibr 251 . 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 7835 1 (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cun 3908  c0 4282  cop 4592  {copab 5167  cmpt 5188   × cxp 5631  Oncon0 6317  suc csuc 6319  cfv 6496  (class class class)co 7357  ωcom 7802  1st c1st 7919  2nd c2nd 7920  reccrdg 8355  1oc1o 8405  2oc2o 8406  𝑔cgoe 33927  𝑔cgna 33928  𝑔cgol 33929   Sat csat 33930
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-map 8767  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937
This theorem is referenced by: (None)
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