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Theorem sat1el2xp 35046
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 6892 . . 3 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
21raleqdv 3315 . 2 (𝑥 = ∅ → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3 fveq2 6892 . . 3 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
43raleqdv 3315 . 2 (𝑥 = 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
5 fveq2 6892 . . 3 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
65raleqdv 3315 . 2 (𝑥 = suc 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
7 fveq2 6892 . . 3 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
87raleqdv 3315 . 2 (𝑥 = 𝑁 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
9 eqeq1 2729 . . . . . . . 8 (𝑥 = (1st𝑤) → (𝑥 = (𝑖𝑔𝑗) ↔ (1st𝑤) = (𝑖𝑔𝑗)))
1092rexbidv 3210 . . . . . . 7 (𝑥 = (1st𝑤) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
1110anbi2d 628 . . . . . 6 (𝑥 = (1st𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
12 eqeq1 2729 . . . . . . 7 (𝑧 = (2nd𝑤) → (𝑧 = ∅ ↔ (2nd𝑤) = ∅))
1312anbi1d 629 . . . . . 6 (𝑧 = (2nd𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)) ↔ ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
1411, 13elopabi 8064 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} → ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
15 goel 35014 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1615eqeq2d 2736 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) ↔ (1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
17 omex 9666 . . . . . . . . . . 11 ω ∈ V
1817, 17pm3.2i 469 . . . . . . . . . 10 (ω ∈ V ∧ ω ∈ V)
19 peano1 7892 . . . . . . . . . . . 12 ∅ ∈ ω
2019a1i 11 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ ω)
21 opelxpi 5709 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
2220, 21opelxpd 5711 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω)))
23 xpeq12 5697 . . . . . . . . . . . . 13 ((𝑎 = ω ∧ 𝑏 = ω) → (𝑎 × 𝑏) = (ω × ω))
2423xpeq2d 5702 . . . . . . . . . . . 12 ((𝑎 = ω ∧ 𝑏 = ω) → (ω × (𝑎 × 𝑏)) = (ω × (ω × ω)))
2524eleq2d 2811 . . . . . . . . . . 11 ((𝑎 = ω ∧ 𝑏 = ω) → (⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω))))
2625spc2egv 3578 . . . . . . . . . 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 2813 . . . . . . . . . 10 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
29282exbidv 1919 . . . . . . . . 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 3190 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3332adantl 480 . . . . 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 35041 . . . 4 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
3634, 35eleq2s 2843 . . 3 (𝑤 ∈ ((∅ Sat ∅)‘∅) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3736rgen 3053 . 2 𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))
38 omsucelsucb 8477 . . . . . . . . . . 11 (𝑦 ∈ ω ↔ suc 𝑦 ∈ suc ω)
39 satf0sucom 35040 . . . . . . . . . . 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 479 . . . . . . . . 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 7874 . . . . . . . . . . . 12 (𝑦 ∈ ω → 𝑦 ∈ On)
43 rdgsuc 8443 . . . . . . . . . . . 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 479 . . . . . . . . . 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 6437 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ suc ω)
47 satf0sucom 35040 . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . 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 6896 . . . . . . . . . . 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 479 . . . . . . . . . 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 2726 . . . . . . . . . . 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 3311 . . . . . . . . . . . . . . . . 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 3324 . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . 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 5209 . . . . . . . . . . . . 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 4157 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 6907 . . . . . . . . . . 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 35042 . . . . . . . . . . . . 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 7749 . . . . . . . . . . . 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 582 . . . . . . . . . . 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 7007 . . . . . . . . . 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 2769 . . . . . . . . 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 2765 . . . . . . . 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 2811 . . . . . . 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 4141 . . . . . . 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 6892 . . . . . . . . . . 11 (𝑤 = 𝑡 → (1st𝑤) = (1st𝑡))
7473eleq1d 2810 . . . . . . . . . 10 (𝑤 = 𝑡 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
75742exbidv 1919 . . . . . . . . 9 (𝑤 = 𝑡 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7675rspccv 3598 . . . . . . . 8 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7776adantl 480 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
78 fveq2 6892 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 → (1st𝑤) = (1st𝑣))
7978eleq1d 2810 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
80792exbidv 1919 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑣 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
8180rspcva 3599 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏)))
82 sels 5434 . . . . . . . . . . . . . . . . . 18 ((1st𝑣) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑣) ∈ 𝑠)
8382exlimivv 1927 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . 15 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑠(1st𝑣) ∈ 𝑠))
86 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (1st𝑤) = (1st𝑢))
8786eleq1d 2810 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
88872exbidv 1919 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
8988rspcva 3599 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏)))
90 sels 5434 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑢) ∈ 𝑠)
9190exlimivv 1927 . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑟 → ((1st𝑢) ∈ 𝑠 ↔ (1st𝑢) ∈ 𝑟))
9493cbvexvw 2032 . . . . . . . . . . . . . . . . . . 19 (∃𝑠(1st𝑢) ∈ 𝑠 ↔ ∃𝑟(1st𝑢) ∈ 𝑟)
95 vex 3467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑟 ∈ V
96 vex 3467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑠 ∈ V
9795, 96pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑟 ∈ V ∧ 𝑠 ∈ V)
98 df-ov 7419 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st𝑢)⊼𝑔(1st𝑣)) = (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩)
99 df-gona 35008 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑔 = (𝑒 ∈ (V × V) ↦ ⟨1o, 𝑒⟩)
100 opeq2 4870 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = ⟨(1st𝑢), (1st𝑣)⟩ → ⟨1o, 𝑒⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
101 opelvvg 5713 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (V × V))
102 opex 5460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V)
10499, 100, 101, 103fvmptd3 7023 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
10598, 104eqtrid 2777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
106 1onn 8659 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1o ∈ ω
107106a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → 1o ∈ ω)
108 opelxpi 5709 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (𝑟 × 𝑠))
109107, 108opelxpd 5711 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ (ω × (𝑟 × 𝑠)))
110105, 109eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠)))
111 xpeq12 5697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 = 𝑟𝑏 = 𝑠) → (𝑎 × 𝑏) = (𝑟 × 𝑠))
112111xpeq2d 5702 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 = 𝑟𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (𝑟 × 𝑠)))
113112eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 = 𝑟𝑏 = 𝑠) → (((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠))))
114113spc2egv 3578 . . . . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
1171162exbidv 1919 . . . . . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑢) ∈ 𝑟 → ((1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
120119exlimdv 1928 . . . . . . . . . . . . . . . . . . . . 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 1925 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 3143 . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . . . . . 21 (ω ∈ V ∧ 𝑠 ∈ V)
132 df-goal 35009 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑔𝑖(1st𝑢) = ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩
133 2onn 8661 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2o ∈ ω
134133a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → 2o ∈ ω)
135 opelxpi 5709 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ ω ∧ (1st𝑢) ∈ 𝑠) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
136135ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
137134, 136opelxpd 5711 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ ∈ (ω × (ω × 𝑠)))
138132, 137eqeltrid 2829 . . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → (1st𝑡) ∈ (ω × (ω × 𝑠)))
143 xpeq12 5697 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (𝑎 × 𝑏) = (ω × 𝑠))
144143xpeq2d 5702 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (ω × 𝑠)))
145144eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = ω ∧ 𝑏 = 𝑠) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (ω × 𝑠))))
146145spc2egv 3578 . . . . . . . . . . . . . . . . . . . . 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 1925 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . 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 3143 . . . . . . . . . . 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 3138 . . . . . . . . 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 480 . . . . . . . 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 2729 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ (1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
161160rexbidv 3169 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)(1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣))))
162 eqeq1 2729 . . . . . . . . . . . . 13 (𝑥 = (1st𝑡) → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
163162rexbidv 3169 . . . . . . . . . . . 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 3169 . . . . . . . . . 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 628 . . . . . . . . 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 2729 . . . . . . . . . 10 (𝑧 = (2nd𝑡) → (𝑧 = ∅ ↔ (2nd𝑡) = ∅))
168167anbi1d 629 . . . . . . . . 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 8064 . . . . . . . 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 411 . . . 4 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
174173ralrimdv 3142 . . 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, 175imbitrrdi 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 7902 1 (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3051  wrex 3060  Vcvv 3463  cun 3937  c0 4318  cop 4630  {copab 5205  cmpt 5226   × cxp 5670  Oncon0 6364  suc csuc 6366  cfv 6543  (class class class)co 7416  ωcom 7868  1st c1st 7989  2nd c2nd 7990  reccrdg 8428  1oc1o 8478  2oc2o 8479  𝑔cgoe 35000  𝑔cgna 35001  𝑔cgol 35002   Sat csat 35003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-map 8845  df-goel 35007  df-gona 35008  df-goal 35009  df-sat 35010
This theorem is referenced by: (None)
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