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Theorem sat1el2xp 35385
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 6905 . . 3 (𝑥 = ∅ → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘∅))
21raleqdv 3325 . 2 (𝑥 = ∅ → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3 fveq2 6905 . . 3 (𝑥 = 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑦))
43raleqdv 3325 . 2 (𝑥 = 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
5 fveq2 6905 . . 3 (𝑥 = suc 𝑦 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘suc 𝑦))
65raleqdv 3325 . 2 (𝑥 = suc 𝑦 → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑥)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
7 fveq2 6905 . . 3 (𝑥 = 𝑁 → ((∅ Sat ∅)‘𝑥) = ((∅ Sat ∅)‘𝑁))
87raleqdv 3325 . 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 3221 . . . . . . 7 (𝑥 = (1st𝑤) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
1110anbi2d 630 . . . . . 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 631 . . . . . 6 (𝑧 = (2nd𝑤) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)) ↔ ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗))))
1411, 13elopabi 8088 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} → ((2nd𝑤) = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗)))
15 goel 35353 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1615eqeq2d 2747 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) ↔ (1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
17 omex 9684 . . . . . . . . . . 11 ω ∈ V
1817, 17pm3.2i 470 . . . . . . . . . 10 (ω ∈ V ∧ ω ∈ V)
19 peano1 7911 . . . . . . . . . . . 12 ∅ ∈ ω
2019a1i 11 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ ω)
21 opelxpi 5721 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨𝑖, 𝑗⟩ ∈ (ω × ω))
2220, 21opelxpd 5723 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω)))
23 xpeq12 5709 . . . . . . . . . . . . 13 ((𝑎 = ω ∧ 𝑏 = ω) → (𝑎 × 𝑏) = (ω × ω))
2423xpeq2d 5714 . . . . . . . . . . . 12 ((𝑎 = ω ∧ 𝑏 = ω) → (ω × (𝑎 × 𝑏)) = (ω × (ω × ω)))
2524eleq2d 2826 . . . . . . . . . . 11 ((𝑎 = ω ∧ 𝑏 = ω) → (⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (ω × ω))))
2625spc2egv 3598 . . . . . . . . . 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 2828 . . . . . . . . . 10 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
29282exbidv 1923 . . . . . . . . 9 ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏⟨∅, ⟨𝑖, 𝑗⟩⟩ ∈ (ω × (𝑎 × 𝑏))))
3027, 29syl5ibrcom 247 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3116, 30sylbid 240 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))))
3231rexlimivv 3200 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (1st𝑤) = (𝑖𝑔𝑗) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3332adantl 481 . . . . 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 35380 . . . 4 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
3634, 35eleq2s 2858 . . 3 (𝑤 ∈ ((∅ Sat ∅)‘∅) → ∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
3736rgen 3062 . 2 𝑤 ∈ ((∅ Sat ∅)‘∅)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))
38 omsucelsucb 8499 . . . . . . . . . . 11 (𝑦 ∈ ω ↔ suc 𝑦 ∈ suc ω)
39 satf0sucom 35379 . . . . . . . . . . 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 217 . . . . . . . . . 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 480 . . . . . . . . 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 7894 . . . . . . . . . . . 12 (𝑦 ∈ ω → 𝑦 ∈ On)
43 rdgsuc 8465 . . . . . . . . . . . 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 480 . . . . . . . . . 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 6456 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ suc ω)
47 satf0sucom 35379 . . . . . . . . . . . . . 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 6909 . . . . . . . . . . 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 480 . . . . . . . . . 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 3321 . . . . . . . . . . . . . . . . 17 (𝑓 = ((∅ Sat ∅)‘𝑦) → (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ ((∅ Sat ∅)‘𝑦)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
5554orbi1d 916 . . . . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 5208 . . . . . . . . . . . . 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 4168 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 6920 . . . . . . . . . . 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 35381 . . . . . . . . . . . . 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 1372 . . . . . . . . . . . 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 7764 . . . . . . . . . . . 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 7022 . . . . . . . . . 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 2826 . . . . . . 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 4152 . . . . . . 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 287 . . . . . 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 6905 . . . . . . . . . . 11 (𝑤 = 𝑡 → (1st𝑤) = (1st𝑡))
7473eleq1d 2825 . . . . . . . . . 10 (𝑤 = 𝑡 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
75742exbidv 1923 . . . . . . . . 9 (𝑤 = 𝑡 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7675rspccv 3618 . . . . . . . 8 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
7776adantl 481 . . . . . . 7 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
78 fveq2 6905 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 → (1st𝑤) = (1st𝑣))
7978eleq1d 2825 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
80792exbidv 1923 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑣 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏))))
8180rspcva 3619 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑣) ∈ (ω × (𝑎 × 𝑏)))
82 sels 5442 . . . . . . . . . . . . . . . . . 18 ((1st𝑣) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑣) ∈ 𝑠)
8382exlimivv 1931 . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 15 (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑣 ∈ ((∅ Sat ∅)‘𝑦) → ∃𝑠(1st𝑣) ∈ 𝑠))
86 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (1st𝑤) = (1st𝑢))
8786eleq1d 2825 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
88872exbidv 1923 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑢 → (∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏))))
8988rspcva 3619 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∈ ((∅ Sat ∅)‘𝑦) ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → ∃𝑎𝑏(1st𝑢) ∈ (ω × (𝑎 × 𝑏)))
90 sels 5442 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) ∈ (ω × (𝑎 × 𝑏)) → ∃𝑠(1st𝑢) ∈ 𝑠)
9190exlimivv 1931 . . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑟 → ((1st𝑢) ∈ 𝑠 ↔ (1st𝑢) ∈ 𝑟))
9493cbvexvw 2035 . . . . . . . . . . . . . . . . . . 19 (∃𝑠(1st𝑢) ∈ 𝑠 ↔ ∃𝑟(1st𝑢) ∈ 𝑟)
95 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑟 ∈ V
96 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑠 ∈ V
9795, 96pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑟 ∈ V ∧ 𝑠 ∈ V)
98 df-ov 7435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st𝑢)⊼𝑔(1st𝑣)) = (⊼𝑔‘⟨(1st𝑢), (1st𝑣)⟩)
99 df-gona 35347 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑔 = (𝑒 ∈ (V × V) ↦ ⟨1o, 𝑒⟩)
100 opeq2 4873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = ⟨(1st𝑢), (1st𝑣)⟩ → ⟨1o, 𝑒⟩ = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
101 opelvvg 5725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (V × V))
102 opex 5468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ V)
10499, 100, 101, 103fvmptd3 7038 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8679 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1o ∈ ω
107106a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → 1o ∈ ω)
108 opelxpi 5721 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨(1st𝑢), (1st𝑣)⟩ ∈ (𝑟 × 𝑠))
109107, 108opelxpd 5723 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ ∈ (ω × (𝑟 × 𝑠)))
110105, 109eqeltrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠)))
111 xpeq12 5709 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 = 𝑟𝑏 = 𝑠) → (𝑎 × 𝑏) = (𝑟 × 𝑠))
112111xpeq2d 5714 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 = 𝑟𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (𝑟 × 𝑠)))
113112eleq2d 2826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 = 𝑟𝑏 = 𝑠) → (((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑟 × 𝑠))))
114113spc2egv 3598 . . . . . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
1171162exbidv 1923 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ ∃𝑎𝑏((1st𝑢)⊼𝑔(1st𝑣)) ∈ (ω × (𝑎 × 𝑏))))
118115, 117syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) ∈ 𝑟 ∧ (1st𝑣) ∈ 𝑠) → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
119118ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑢) ∈ 𝑟 → ((1st𝑣) ∈ 𝑠 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
120119exlimdv 1932 . . . . . . . . . . . . . . . . . . . . 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 1929 . . . . . . . . . . . . . . . . . . 19 (∃𝑟(1st𝑢) ∈ 𝑟 → ((1st𝑡) = ((1st𝑢)⊼𝑔(1st𝑣)) → (∃𝑠(1st𝑣) ∈ 𝑠 → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
12394, 122sylbi 217 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 3152 . . . . . . . . . . 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 470 . . . . . . . . . . . . . . . . . . . . 21 (ω ∈ V ∧ 𝑠 ∈ V)
132 df-goal 35348 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑔𝑖(1st𝑢) = ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩
133 2onn 8681 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2o ∈ ω
134133a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → 2o ∈ ω)
135 opelxpi 5721 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ ω ∧ (1st𝑢) ∈ 𝑠) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
136135ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨𝑖, (1st𝑢)⟩ ∈ (ω × 𝑠))
137134, 136opelxpd 5723 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω) → ⟨2o, ⟨𝑖, (1st𝑢)⟩⟩ ∈ (ω × (ω × 𝑠)))
138132, 137eqeltrid 2844 . . . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) ∈ 𝑠𝑖 ∈ ω ∧ (1st𝑡) = ∀𝑔𝑖(1st𝑢)) → (1st𝑡) ∈ (ω × (ω × 𝑠)))
143 xpeq12 5709 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (𝑎 × 𝑏) = (ω × 𝑠))
144143xpeq2d 5714 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = ω ∧ 𝑏 = 𝑠) → (ω × (𝑎 × 𝑏)) = (ω × (ω × 𝑠)))
145144eleq2d 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = ω ∧ 𝑏 = 𝑠) → ((1st𝑡) ∈ (ω × (𝑎 × 𝑏)) ↔ (1st𝑡) ∈ (ω × (ω × 𝑠))))
146145spc2egv 3598 . . . . . . . . . . . . . . . . . . . . 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 1929 . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 3152 . . . . . . . . . . 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 859 . . . . . . . . . 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 3147 . . . . . . . . 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 481 . . . . . . . 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 3178 . . . . . . . . . . . 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 3178 . . . . . . . . . . . 12 (𝑥 = (1st𝑡) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω (1st𝑡) = ∀𝑔𝑖(1st𝑢)))
164161, 163orbi12d 918 . . . . . . . . . . 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 3178 . . . . . . . . . 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 630 . . . . . . . . 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 631 . . . . . . . . 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 8088 . . . . . . . 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 859 . . . . . 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 240 . . . . 5 ((𝑦 ∈ ω ∧ ∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏))) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
173172ex 412 . . . 4 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → (𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦) → ∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))))
174173ralrimdv 3151 . . 3 (𝑦 ∈ ω → (∀𝑤 ∈ ((∅ Sat ∅)‘𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) → ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏))))
17575cbvralvw 3236 . . 3 (∀𝑤 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)) ↔ ∀𝑡 ∈ ((∅ Sat ∅)‘suc 𝑦)∃𝑎𝑏(1st𝑡) ∈ (ω × (𝑎 × 𝑏)))
176174, 175imbitrrdi 252 . 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 7919 1 (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cun 3948  c0 4332  cop 4631  {copab 5204  cmpt 5224   × cxp 5682  Oncon0 6383  suc csuc 6385  cfv 6560  (class class class)co 7432  ωcom 7888  1st c1st 8013  2nd c2nd 8014  reccrdg 8450  1oc1o 8500  2oc2o 8501  𝑔cgoe 35339  𝑔cgna 35340  𝑔cgol 35341   Sat csat 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-map 8869  df-goel 35346  df-gona 35347  df-goal 35348  df-sat 35349
This theorem is referenced by: (None)
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