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Theorem tx1stc 23001
Description: The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
tx1stc ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ 1stω)

Proof of Theorem tx1stc
Dummy variables 𝑎 𝑏 𝑚 𝑛 𝑝 𝑞 𝑟 𝑠 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stctop 22794 . . 3 (𝑅 ∈ 1stω → 𝑅 ∈ Top)
2 1stctop 22794 . . 3 (𝑆 ∈ 1stω → 𝑆 ∈ Top)
3 txtop 22920 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 596 . 2 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2736 . . . . . . . 8 𝑅 = 𝑅
651stcclb 22795 . . . . . . 7 ((𝑅 ∈ 1stω ∧ 𝑢 𝑅) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
76ad2ant2r 745 . . . . . 6 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
8 eqid 2736 . . . . . . . 8 𝑆 = 𝑆
981stcclb 22795 . . . . . . 7 ((𝑆 ∈ 1stω ∧ 𝑣 𝑆) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
109ad2ant2l 744 . . . . . 6 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
11 reeanv 3217 . . . . . . 7 (∃𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆((𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ↔ (∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))))
12 an4 654 . . . . . . . . 9 (((𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ↔ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))))
13 txopn 22953 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑚𝑅𝑛𝑆)) → (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1413ralrimivva 3197 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
151, 2, 14syl2an 596 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1615adantr 481 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
17 elpwi 4567 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ 𝒫 𝑅𝑎𝑅)
18 ssralv 4010 . . . . . . . . . . . . . . . . . 18 (𝑎𝑅 → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
1917, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ 𝒫 𝑅 → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
20 elpwi 4567 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ 𝒫 𝑆𝑏𝑆)
21 ssralv 4010 . . . . . . . . . . . . . . . . . . 19 (𝑏𝑆 → (∀𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2220, 21syl 17 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ 𝒫 𝑆 → (∀𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2322ralimdv 3166 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ 𝒫 𝑆 → (∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2419, 23sylan9 508 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆) → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2516, 24mpan9 507 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
26 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
2726fmpo 8000 . . . . . . . . . . . . . . 15 (∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
2825, 27sylib 217 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
2928frnd 6676 . . . . . . . . . . . . 13 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
30 ovex 7390 . . . . . . . . . . . . . 14 (𝑅 ×t 𝑆) ∈ V
3130elpw2 5302 . . . . . . . . . . . . 13 (ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆) ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
3229, 31sylibr 233 . . . . . . . . . . . 12 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆))
3332adantr 481 . . . . . . . . . . 11 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆))
34 omelon 9582 . . . . . . . . . . . . . . 15 ω ∈ On
35 xpct 9952 . . . . . . . . . . . . . . 15 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ ω)
36 ondomen 9973 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝑎 × 𝑏) ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
3734, 35, 36sylancr 587 . . . . . . . . . . . . . 14 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
38 vex 3449 . . . . . . . . . . . . . . . . 17 𝑚 ∈ V
39 vex 3449 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4038, 39xpex 7687 . . . . . . . . . . . . . . . 16 (𝑚 × 𝑛) ∈ V
4126, 40fnmpoi 8002 . . . . . . . . . . . . . . 15 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏)
42 dffn4 6762 . . . . . . . . . . . . . . 15 ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)))
4341, 42mpbi 229 . . . . . . . . . . . . . 14 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
44 fodomnum 9993 . . . . . . . . . . . . . 14 ((𝑎 × 𝑏) ∈ dom card → ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏)))
4537, 43, 44mpisyl 21 . . . . . . . . . . . . 13 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏))
46 domtr 8947 . . . . . . . . . . . . 13 ((ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
4745, 35, 46syl2anc 584 . . . . . . . . . . . 12 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
4847ad2antrl 726 . . . . . . . . . . 11 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
491, 2anim12i 613 . . . . . . . . . . . . . . 15 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ∈ Top ∧ 𝑆 ∈ Top))
5049ad3antrrr 728 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → (𝑅 ∈ Top ∧ 𝑆 ∈ Top))
51 eltx 22919 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑧 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
5250, 51syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → (𝑧 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
53 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑢, 𝑣⟩ → (𝑤 ∈ (𝑟 × 𝑠) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠)))
5453anbi1d 630 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
55542rexbidv 3213 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑢, 𝑣⟩ → (∃𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
5655rspccv 3578 . . . . . . . . . . . . . 14 (∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
57 r19.27v 3184 . . . . . . . . . . . . . . . . . 18 ((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
58 r19.29 3117 . . . . . . . . . . . . . . . . . . 19 ((∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑟𝑅 (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
59 r19.29 3117 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
60 opelxp 5669 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ↔ (𝑢𝑟𝑣𝑠))
61 pm3.35 801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))
62 pm3.35 801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))
6361, 62anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6463an4s 658 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑟𝑣𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6560, 64sylanb 581 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6665anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6766anasss 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6867an12s 647 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6968expl 458 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7069reximdv 3167 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7159, 70syl5 34 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → ((∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7271impl 456 . . . . . . . . . . . . . . . . . . . 20 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7372reximi 3087 . . . . . . . . . . . . . . . . . . 19 (∃𝑟𝑅 (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑟𝑅𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7458, 73syl 17 . . . . . . . . . . . . . . . . . 18 ((∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑟𝑅𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7557, 74sylan 580 . . . . . . . . . . . . . . . . 17 (((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑟𝑅𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
76 reeanv 3217 . . . . . . . . . . . . . . . . . . . 20 (∃𝑝𝑎𝑞𝑏 ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ↔ (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
77 simpr1l 1230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → 𝑝𝑎)
78 simpr1r 1231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → 𝑞𝑏)
79 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑝 × 𝑞) = (𝑝 × 𝑞))
80 xpeq1 5647 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑝 → (𝑚 × 𝑛) = (𝑝 × 𝑛))
8180eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑝 → ((𝑝 × 𝑞) = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑛)))
82 xpeq2 5654 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑞 → (𝑝 × 𝑛) = (𝑝 × 𝑞))
8382eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑞 → ((𝑝 × 𝑞) = (𝑝 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑞)))
8481, 83rspc2ev 3592 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑎𝑞𝑏 ∧ (𝑝 × 𝑞) = (𝑝 × 𝑞)) → ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
8577, 78, 79, 84syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
86 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑝 ∈ V
87 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑞 ∈ V
8886, 87xpex 7687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 × 𝑞) ∈ V
89 eqeq1 2740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝑝 × 𝑞) → (𝑥 = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑚 × 𝑛)))
90892rexbidv 3213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑝 × 𝑞) → (∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛) ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛)))
9188, 90elab 3630 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝 × 𝑞) ∈ {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)} ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
9285, 91sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑝 × 𝑞) ∈ {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)})
9326rnmpo 7489 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)}
9492, 93eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑝 × 𝑞) ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)))
95 simpr2 1195 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)))
96 opelxpi 5670 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢𝑝𝑣𝑞) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
9796ad2ant2r 745 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
9895, 97syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
99 xpss12 5648 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑟𝑞𝑠) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
10099ad2ant2l 744 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
10195, 100syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
102 simpr3 1196 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑟 × 𝑠) ⊆ 𝑧)
103101, 102sstrd 3954 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → (𝑝 × 𝑞) ⊆ 𝑧)
104 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (⟨𝑢, 𝑣⟩ ∈ 𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞)))
105 sseq1 3969 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (𝑤𝑧 ↔ (𝑝 × 𝑞) ⊆ 𝑧))
106104, 105anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑝 × 𝑞) → ((⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)))
107106rspcev 3581 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝 × 𝑞) ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))
10894, 98, 103, 107syl12anc 835 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) ∧ ((𝑝𝑎𝑞𝑏) ∧ ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))
1091083exp2 1354 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) → ((𝑝𝑎𝑞𝑏) → (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → ((𝑟 × 𝑠) ⊆ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
110109rexlimdvv 3204 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) → (∃𝑝𝑎𝑞𝑏 ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → ((𝑟 × 𝑠) ⊆ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
11176, 110biimtrrid 242 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) → ((𝑟 × 𝑠) ⊆ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
112111impd 411 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) ∧ (𝑟𝑅𝑠𝑆)) → (((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
113112rexlimdvva 3205 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) → (∃𝑟𝑅𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
11475, 113syl5 34 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) → (((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
115114expd 416 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ (𝑎 ≼ ω ∧ 𝑏 ≼ ω)) → ((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → (∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
116115impr 455 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → (∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
11756, 116syl9r 78 . . . . . . . . . . . . 13 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → (∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
11852, 117sylbid 239 . . . . . . . . . . . 12 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → (𝑧 ∈ (𝑅 ×t 𝑆) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
119118ralrimiv 3142 . . . . . . . . . . 11 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
120 breq1 5108 . . . . . . . . . . . . 13 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (𝑦 ≼ ω ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω))
121 rexeq 3310 . . . . . . . . . . . . . . 15 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
122121imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → ((⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
123122ralbidv 3174 . . . . . . . . . . . . 13 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
124120, 123anbi12d 631 . . . . . . . . . . . 12 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))) ↔ (ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
125124rspcev 3581 . . . . . . . . . . 11 ((ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
12633, 48, 119, 125syl12anc 835 . . . . . . . . . 10 (((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) ∧ ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
127126ex 413 . . . . . . . . 9 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → (((𝑎 ≼ ω ∧ 𝑏 ≼ ω) ∧ (∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
12812, 127biimtrid 241 . . . . . . . 8 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → (((𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
129128rexlimdvva 3205 . . . . . . 7 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → (∃𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆((𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
13011, 129biimtrrid 242 . . . . . 6 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ((∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
1317, 10, 130mp2and 697 . . . . 5 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
132131ralrimivva 3197 . . . 4 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ∀𝑢 𝑅𝑣 𝑆𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
133 eleq1 2825 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑧 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑧))
134 eleq1 2825 . . . . . . . . . . 11 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑤))
135134anbi1d 630 . . . . . . . . . 10 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
136135rexbidv 3175 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
137133, 136imbi12d 344 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
138137ralbidv 3174 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → (∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
139138anbi2d 629 . . . . . 6 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
140139rexbidv 3175 . . . . 5 (𝑥 = ⟨𝑢, 𝑣⟩ → (∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))))
141140ralxp 5797 . . . 4 (∀𝑥 ∈ ( 𝑅 × 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ∀𝑢 𝑅𝑣 𝑆𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
142132, 141sylibr 233 . . 3 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ∀𝑥 ∈ ( 𝑅 × 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1435, 8txuni 22943 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1441, 2, 143syl2an 596 . . . 4 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
145144raleqdv 3313 . . 3 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (∀𝑥 ∈ ( 𝑅 × 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ∀𝑥 (𝑅 ×t 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
146142, 145mpbid 231 . 2 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ∀𝑥 (𝑅 ×t 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
147 eqid 2736 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
148147is1stc2 22793 . 2 ((𝑅 ×t 𝑆) ∈ 1stω ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 (𝑅 ×t 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
1494, 146, 148sylanbrc 583 1 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ 1stω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  wss 3910  𝒫 cpw 4560  cop 4592   cuni 4865   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  Oncon0 6317   Fn wfn 6491  wf 6492  ontowfo 6494  (class class class)co 7357  cmpo 7359  ωcom 7802  cdom 8881  cardccrd 9871  Topctop 22242  1stωc1stc 22788   ×t ctx 22911
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-rmo 3353  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-int 4908  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-se 5589  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-isom 6505  df-riota 7313  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-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-card 9875  df-acn 9878  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-1stc 22790  df-tx 22913
This theorem is referenced by: (None)
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