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Theorem tx1stc 21263
 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 21056 . . 3 (𝑅 ∈ 1st𝜔 → 𝑅 ∈ Top)
2 1stctop 21056 . . 3 (𝑆 ∈ 1st𝜔 → 𝑆 ∈ Top)
3 txtop 21182 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 493 . 2 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2610 . . . . . . . 8 𝑅 = 𝑅
651stcclb 21057 . . . . . . 7 ((𝑅 ∈ 1st𝜔 ∧ 𝑢 𝑅) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
76ad2ant2r 779 . . . . . 6 (((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
8 eqid 2610 . . . . . . . 8 𝑆 = 𝑆
981stcclb 21057 . . . . . . 7 ((𝑆 ∈ 1st𝜔 ∧ 𝑣 𝑆) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
109ad2ant2l 778 . . . . . 6 (((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
11 reeanv 3086 . . . . . . 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 861 . . . . . . . . 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 21215 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑚𝑅𝑛𝑆)) → (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1413ralrimivva 2954 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
151, 2, 14syl2an 493 . . . . . . . . . . . . . . . . 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 4117 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ 𝒫 𝑅𝑎𝑅)
18 ssralv 3629 . . . . . . . . . . . . . . . . . 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 4117 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ 𝒫 𝑆𝑏𝑆)
21 ssralv 3629 . . . . . . . . . . . . . . . . . . 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 2946 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ 𝒫 𝑆 → (∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2419, 23sylan9 687 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆) → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2516, 24mpan9 485 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
26 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
2726fmpt2 7126 . . . . . . . . . . . . . . 15 (∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
2825, 27sylib 207 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
29 frn 5966 . . . . . . . . . . . . . 14 ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
3028, 29syl 17 . . . . . . . . . . . . 13 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
31 ovex 6577 . . . . . . . . . . . . . 14 (𝑅 ×t 𝑆) ∈ V
3231elpw2 4755 . . . . . . . . . . . . 13 (ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆) ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
3330, 32sylibr 223 . . . . . . . . . . . 12 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆))
3433adantr 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))
35 omelon 8426 . . . . . . . . . . . . . . 15 ω ∈ On
36 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ V
3736xpdom1 7944 . . . . . . . . . . . . . . . . 17 (𝑎 ≼ ω → (𝑎 × 𝑏) ≼ (ω × 𝑏))
38 omex 8423 . . . . . . . . . . . . . . . . . 18 ω ∈ V
3938xpdom2 7940 . . . . . . . . . . . . . . . . 17 (𝑏 ≼ ω → (ω × 𝑏) ≼ (ω × ω))
40 domtr 7895 . . . . . . . . . . . . . . . . 17 (((𝑎 × 𝑏) ≼ (ω × 𝑏) ∧ (ω × 𝑏) ≼ (ω × ω)) → (𝑎 × 𝑏) ≼ (ω × ω))
4137, 39, 40syl2an 493 . . . . . . . . . . . . . . . 16 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ (ω × ω))
42 xpomen 8721 . . . . . . . . . . . . . . . 16 (ω × ω) ≈ ω
43 domentr 7901 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑏) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑎 × 𝑏) ≼ ω)
4441, 42, 43sylancl 693 . . . . . . . . . . . . . . 15 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ ω)
45 ondomen 8743 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝑎 × 𝑏) ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
4635, 44, 45sylancr 694 . . . . . . . . . . . . . 14 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
47 vex 3176 . . . . . . . . . . . . . . . . 17 𝑚 ∈ V
48 vex 3176 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4947, 48xpex 6860 . . . . . . . . . . . . . . . 16 (𝑚 × 𝑛) ∈ V
5026, 49fnmpt2i 7128 . . . . . . . . . . . . . . 15 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏)
51 dffn4 6034 . . . . . . . . . . . . . . 15 ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)))
5250, 51mpbi 219 . . . . . . . . . . . . . 14 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
53 fodomnum 8763 . . . . . . . . . . . . . 14 ((𝑎 × 𝑏) ∈ dom card → ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏)))
5446, 52, 53mpisyl 21 . . . . . . . . . . . . 13 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏))
55 domtr 7895 . . . . . . . . . . . . 13 ((ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
5654, 44, 55syl2anc 691 . . . . . . . . . . . 12 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
5756ad2antrl 760 . . . . . . . . . . 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)
581, 2anim12i 588 . . . . . . . . . . . . . . 15 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ∈ Top ∧ 𝑆 ∈ Top))
5958ad3antrrr 762 . . . . . . . . . . . . . 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))
60 eltx 21181 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑧 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
6159, 60syl 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)))
62 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑢, 𝑣⟩ → (𝑤 ∈ (𝑟 × 𝑠) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠)))
6362anbi1d 737 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
64632rexbidv 3039 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑢, 𝑣⟩ → (∃𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
6564rspccv 3279 . . . . . . . . . . . . . 14 (∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
66 r19.27v 3052 . . . . . . . . . . . . . . . . . 18 ((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
67 r19.29 3054 . . . . . . . . . . . . . . . . . . 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)))
68 r19.29 3054 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
69 opelxp 5070 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ↔ (𝑢𝑟𝑣𝑠))
70 pm3.35 609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))
71 pm3.35 609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))
7270, 71anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7372an4s 865 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑟𝑣𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7469, 73sylanb 488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7574anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7675anasss 677 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7776an12s 839 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7877expl 646 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7978reximdv 2999 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
8068, 79syl5 33 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → ((∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
8180impl 648 . . . . . . . . . . . . . . . . . . . 20 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
8281reximi 2994 . . . . . . . . . . . . . . . . . . 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))
8367, 82syl 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))
8466, 83sylan 487 . . . . . . . . . . . . . . . . 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))
85 reeanv 3086 . . . . . . . . . . . . . . . . . . . 20 (∃𝑝𝑎𝑞𝑏 ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ↔ (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
86 simpr1l 1111 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
87 simpr1r 1112 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
88 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
89 xpeq1 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑝 → (𝑚 × 𝑛) = (𝑝 × 𝑛))
9089eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑝 → ((𝑝 × 𝑞) = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑛)))
91 xpeq2 5053 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑞 → (𝑝 × 𝑛) = (𝑝 × 𝑞))
9291eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑞 → ((𝑝 × 𝑞) = (𝑝 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑞)))
9390, 92rspc2ev 3295 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑎𝑞𝑏 ∧ (𝑝 × 𝑞) = (𝑝 × 𝑞)) → ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
9486, 87, 88, 93syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . 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))
95 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑝 ∈ V
96 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑞 ∈ V
9795, 96xpex 6860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 × 𝑞) ∈ V
98 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝑝 × 𝑞) → (𝑥 = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑚 × 𝑛)))
99982rexbidv 3039 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑝 × 𝑞) → (∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛) ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛)))
10097, 99elab 3319 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝 × 𝑞) ∈ {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)} ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
10194, 100sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . 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)})
10226rnmpt2 6668 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)}
103101, 102syl6eleqr 2699 . . . . . . . . . . . . . . . . . . . . . . 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)))
104 simpr2 1061 . . . . . . . . . . . . . . . . . . . . . . . 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)))
105 opelxpi 5072 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢𝑝𝑣𝑞) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
106105ad2ant2r 779 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
107104, 106syl 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))
108 xpss12 5148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑟𝑞𝑠) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
109108ad2ant2l 778 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
110104, 109syl 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))
111 simpr3 1062 . . . . . . . . . . . . . . . . . . . . . . . 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)
112110, 111sstrd 3578 . . . . . . . . . . . . . . . . . . . . . . 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)
113 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (⟨𝑢, 𝑣⟩ ∈ 𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞)))
114 sseq1 3589 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (𝑤𝑧 ↔ (𝑝 × 𝑞) ⊆ 𝑧))
115113, 114anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑝 × 𝑞) → ((⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)))
116115rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝 × 𝑞) ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))
117103, 107, 112, 116syl12anc 1316 . . . . . . . . . . . . . . . . . . . . . 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))
1181173exp2 1277 . . . . . . . . . . . . . . . . . . . . 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)))))
119118rexlimdvv 3019 . . . . . . . . . . . . . . . . . . . 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))))
12085, 119syl5bir 232 . . . . . . . . . . . . . . . . . . 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))))
121120impd 446 . . . . . . . . . . . . . . . . . 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)))
122121rexlimdvva 3020 . . . . . . . . . . . . . . . . 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)))
12384, 122syl5 33 . . . . . . . . . . . . . . . 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)))
124123expd 451 . . . . . . . . . . . . . . 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))))
125124impr 647 . . . . . . . . . . . . . 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)))
12665, 125syl9r 76 . . . . . . . . . . . . 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))))
12761, 126sylbid 229 . . . . . . . . . . . 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))))
128127ralrimiv 2948 . . . . . . . . . . 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)))
129 breq1 4586 . . . . . . . . . . . . 13 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (𝑦 ≼ ω ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω))
130 rexeq 3116 . . . . . . . . . . . . . . 15 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
131130imbi2d 329 . . . . . . . . . . . . . 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))))
132131ralbidv 2969 . . . . . . . . . . . . 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))))
133129, 132anbi12d 743 . . . . . . . . . . . 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)))))
134133rspcev 3282 . . . . . . . . . . 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))))
13534, 57, 128, 134syl12anc 1316 . . . . . . . . . 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))))
136135ex 449 . . . . . . . . 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)))))
13712, 136syl5bi 231 . . . . . . . 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)))))
138137rexlimdvva 3020 . . . . . . 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)))))
13911, 138syl5bir 232 . . . . . 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)))))
1407, 10, 139mp2and 711 . . . . 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))))
141140ralrimivva 2954 . . . 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))))
142 eleq1 2676 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑧 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑧))
143 eleq1 2676 . . . . . . . . . . 11 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑤))
144143anbi1d 737 . . . . . . . . . 10 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
145144rexbidv 3034 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
146142, 145imbi12d 333 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
147146ralbidv 2969 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → (∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
148147anbi2d 736 . . . . . 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)))))
149148rexbidv 3034 . . . . 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)))))
150149ralxp 5185 . . . 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))))
151141, 150sylibr 223 . . 3 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ∀𝑥 ∈ ( 𝑅 × 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1525, 8txuni 21205 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1531, 2, 152syl2an 493 . . . 4 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
154153raleqdv 3121 . . 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)))))
155151, 154mpbid 221 . 2 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ∀𝑥 (𝑅 ×t 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
156 eqid 2610 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
157156is1stc2 21055 . 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)))))
1584, 155, 157sylanbrc 695 1 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ×t 𝑆) ∈ 1st𝜔)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ≈ cen 7838   ≼ cdom 7839  cardccrd 8644  Topctop 20517  1st𝜔c1stc 21050   ×t ctx 21173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-acn 8651  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-1stc 21052  df-tx 21175 This theorem is referenced by: (None)
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