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Theorem tx1stc 23478
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 23271 . . 3 (𝑅 ∈ 1stω → 𝑅 ∈ Top)
2 1stctop 23271 . . 3 (𝑆 ∈ 1stω → 𝑆 ∈ Top)
3 txtop 23397 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 595 . 2 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2724 . . . . . . . 8 𝑅 = 𝑅
651stcclb 23272 . . . . . . 7 ((𝑅 ∈ 1stω ∧ 𝑢 𝑅) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
76ad2ant2r 744 . . . . . 6 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
8 eqid 2724 . . . . . . . 8 𝑆 = 𝑆
981stcclb 23272 . . . . . . 7 ((𝑆 ∈ 1stω ∧ 𝑣 𝑆) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
109ad2ant2l 743 . . . . . 6 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
11 reeanv 3218 . . . . . . 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 653 . . . . . . . . 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 23430 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑚𝑅𝑛𝑆)) → (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1413ralrimivva 3192 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
151, 2, 14syl2an 595 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1615adantr 480 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
17 elpwi 4602 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ 𝒫 𝑅𝑎𝑅)
18 ssralv 4043 . . . . . . . . . . . . . . . . . 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 4602 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ 𝒫 𝑆𝑏𝑆)
21 ssralv 4043 . . . . . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ 𝒫 𝑆 → (∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2419, 23sylan9 507 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆) → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2516, 24mpan9 506 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
26 eqid 2724 . . . . . . . . . . . . . . . 16 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
2726fmpo 8048 . . . . . . . . . . . . . . 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 6716 . . . . . . . . . . . . 13 ((((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
30 ovex 7435 . . . . . . . . . . . . . 14 (𝑅 ×t 𝑆) ∈ V
3130elpw2 5336 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 9638 . . . . . . . . . . . . . . 15 ω ∈ On
35 xpct 10008 . . . . . . . . . . . . . . 15 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ ω)
36 ondomen 10029 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝑎 × 𝑏) ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
3734, 35, 36sylancr 586 . . . . . . . . . . . . . 14 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
38 vex 3470 . . . . . . . . . . . . . . . . 17 𝑚 ∈ V
39 vex 3470 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4038, 39xpex 7734 . . . . . . . . . . . . . . . 16 (𝑚 × 𝑛) ∈ V
4126, 40fnmpoi 8050 . . . . . . . . . . . . . . 15 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏)
42 dffn4 6802 . . . . . . . . . . . . . . 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 10049 . . . . . . . . . . . . . 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 9000 . . . . . . . . . . . . 13 ((ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
4745, 35, 46syl2anc 583 . . . . . . . . . . . 12 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
4847ad2antrl 725 . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 15 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ∈ Top ∧ 𝑆 ∈ Top))
5049ad3antrrr 727 . . . . . . . . . . . . . 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 23396 . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑢, 𝑣⟩ → (𝑤 ∈ (𝑟 × 𝑠) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠)))
5453anbi1d 629 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
55542rexbidv 3211 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑢, 𝑣⟩ → (∃𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
5655rspccv 3601 . . . . . . . . . . . . . 14 (∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
57 r19.27v 3179 . . . . . . . . . . . . . . . . . 18 ((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
58 r19.29 3106 . . . . . . . . . . . . . . . . . . 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 3106 . . . . . . . . . . . . . . . . . . . . . 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 5703 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ↔ (𝑢𝑟𝑣𝑠))
61 pm3.35 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))
62 pm3.35 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))
6361, 62anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6463an4s 657 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑟𝑣𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6560, 64sylanb 580 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
6665anim1i 614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6766anasss 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6867an12s 646 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
6968expl 457 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7069reximdv 3162 . . . . . . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . . . . . 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 3076 . . . . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . . . 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 3218 . . . . . . . . . . . . . . . . . . . 20 (∃𝑝𝑎𝑞𝑏 ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ↔ (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
77 simpr1l 1227 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5681 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑝 → (𝑚 × 𝑛) = (𝑝 × 𝑛))
8180eqeq2d 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑝 → ((𝑝 × 𝑞) = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑛)))
82 xpeq2 5688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑞 → (𝑝 × 𝑛) = (𝑝 × 𝑞))
8382eqeq2d 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑞 → ((𝑝 × 𝑞) = (𝑝 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑞)))
8481, 83rspc2ev 3617 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑎𝑞𝑏 ∧ (𝑝 × 𝑞) = (𝑝 × 𝑞)) → ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
8577, 78, 79, 84syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . 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 3470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑝 ∈ V
87 vex 3470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑞 ∈ V
8886, 87xpex 7734 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 × 𝑞) ∈ V
89 eqeq1 2728 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝑝 × 𝑞) → (𝑥 = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑚 × 𝑛)))
90892rexbidv 3211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑝 × 𝑞) → (∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛) ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛)))
9188, 90elab 3661 . . . . . . . . . . . . . . . . . . . . . . . . 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 7535 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)}
9492, 93eleqtrrdi 2836 . . . . . . . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢𝑝𝑣𝑞) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
9796ad2ant2r 744 . . . . . . . . . . . . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑟𝑞𝑠) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
10099ad2ant2l 743 . . . . . . . . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . . . . . . 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 3985 . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (⟨𝑢, 𝑣⟩ ∈ 𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞)))
105 sseq1 4000 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (𝑤𝑧 ↔ (𝑝 × 𝑞) ⊆ 𝑧))
106104, 105anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑝 × 𝑞) → ((⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)))
107106rspcev 3604 . . . . . . . . . . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . 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 3203 . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . 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 3137 . . . . . . . . . . 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 5142 . . . . . . . . . . . . 13 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (𝑦 ≼ ω ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω))
121 rexeq 3313 . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . 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 630 . . . . . . . . . . . 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 3604 . . . . . . . . . . 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 834 . . . . . . . . . 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 412 . . . . . . . . 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 3203 . . . . . . 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 696 . . . . 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 3192 . . . 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 2813 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑧 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑧))
134 eleq1 2813 . . . . . . . . . . 11 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑤))
135134anbi1d 629 . . . . . . . . . 10 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
136135rexbidv 3170 . . . . . . . . 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 3169 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → (∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
139138anbi2d 628 . . . . . 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 3170 . . . . 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 5832 . . . 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 23420 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1441, 2, 143syl2an 595 . . . 4 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
145144raleqdv 3317 . . 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 2724 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
148147is1stc2 23270 . 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 582 1 ((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ 1stω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  {cab 2701  wral 3053  wrex 3062  wss 3941  𝒫 cpw 4595  cop 4627   cuni 4900   class class class wbr 5139   × cxp 5665  dom cdm 5667  ran crn 5668  Oncon0 6355   Fn wfn 6529  wf 6530  ontowfo 6532  (class class class)co 7402  cmpo 7404  ωcom 7849  cdom 8934  cardccrd 9927  Topctop 22719  1stωc1stc 23265   ×t ctx 23388
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-oi 9502  df-card 9931  df-acn 9934  df-topgen 17390  df-top 22720  df-topon 22737  df-bases 22773  df-1stc 23267  df-tx 23390
This theorem is referenced by: (None)
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