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Theorem tx1stc 21862
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 21655 . . 3 (𝑅 ∈ 1st𝜔 → 𝑅 ∈ Top)
2 1stctop 21655 . . 3 (𝑆 ∈ 1st𝜔 → 𝑆 ∈ Top)
3 txtop 21781 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 589 . 2 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2778 . . . . . . . 8 𝑅 = 𝑅
651stcclb 21656 . . . . . . 7 ((𝑅 ∈ 1st𝜔 ∧ 𝑢 𝑅) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
76ad2ant2r 737 . . . . . 6 (((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑎 ∈ 𝒫 𝑅(𝑎 ≼ ω ∧ ∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))))
8 eqid 2778 . . . . . . . 8 𝑆 = 𝑆
981stcclb 21656 . . . . . . 7 ((𝑆 ∈ 1st𝜔 ∧ 𝑣 𝑆) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
109ad2ant2l 736 . . . . . 6 (((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) → ∃𝑏 ∈ 𝒫 𝑆(𝑏 ≼ ω ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
11 reeanv 3293 . . . . . . 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 646 . . . . . . . . 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 21814 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑚𝑅𝑛𝑆)) → (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1413ralrimivva 3153 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
151, 2, 14syl2an 589 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
1615adantr 474 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) → ∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
17 elpwi 4389 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ 𝒫 𝑅𝑎𝑅)
18 ssralv 3885 . . . . . . . . . . . . . . . . . 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 4389 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ 𝒫 𝑆𝑏𝑆)
21 ssralv 3885 . . . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ 𝒫 𝑆 → (∀𝑚𝑎𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2419, 23sylan9 503 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆) → (∀𝑚𝑅𝑛𝑆 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆)))
2516, 24mpan9 502 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆))
26 eqid 2778 . . . . . . . . . . . . . . . 16 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
2726fmpt2 7517 . . . . . . . . . . . . . . 15 (∀𝑚𝑎𝑛𝑏 (𝑚 × 𝑛) ∈ (𝑅 ×t 𝑆) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
2825, 27sylib 210 . . . . . . . . . . . . . 14 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)⟶(𝑅 ×t 𝑆))
2928frnd 6298 . . . . . . . . . . . . 13 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
30 ovex 6954 . . . . . . . . . . . . . 14 (𝑅 ×t 𝑆) ∈ V
3130elpw2 5062 . . . . . . . . . . . . 13 (ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆) ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ⊆ (𝑅 ×t 𝑆))
3229, 31sylibr 226 . . . . . . . . . . . 12 ((((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) ∧ (𝑢 𝑅𝑣 𝑆)) ∧ (𝑎 ∈ 𝒫 𝑅𝑏 ∈ 𝒫 𝑆)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∈ 𝒫 (𝑅 ×t 𝑆))
3332adantr 474 . . . . . . . . . . 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 8840 . . . . . . . . . . . . . . 15 ω ∈ On
35 vex 3401 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ V
3635xpdom1 8347 . . . . . . . . . . . . . . . . 17 (𝑎 ≼ ω → (𝑎 × 𝑏) ≼ (ω × 𝑏))
37 omex 8837 . . . . . . . . . . . . . . . . . 18 ω ∈ V
3837xpdom2 8343 . . . . . . . . . . . . . . . . 17 (𝑏 ≼ ω → (ω × 𝑏) ≼ (ω × ω))
39 domtr 8294 . . . . . . . . . . . . . . . . 17 (((𝑎 × 𝑏) ≼ (ω × 𝑏) ∧ (ω × 𝑏) ≼ (ω × ω)) → (𝑎 × 𝑏) ≼ (ω × ω))
4036, 38, 39syl2an 589 . . . . . . . . . . . . . . . 16 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ (ω × ω))
41 xpomen 9171 . . . . . . . . . . . . . . . 16 (ω × ω) ≈ ω
42 domentr 8300 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑏) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑎 × 𝑏) ≼ ω)
4340, 41, 42sylancl 580 . . . . . . . . . . . . . . 15 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ≼ ω)
44 ondomen 9193 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝑎 × 𝑏) ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
4534, 43, 44sylancr 581 . . . . . . . . . . . . . 14 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → (𝑎 × 𝑏) ∈ dom card)
46 vex 3401 . . . . . . . . . . . . . . . . 17 𝑚 ∈ V
47 vex 3401 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4846, 47xpex 7240 . . . . . . . . . . . . . . . 16 (𝑚 × 𝑛) ∈ V
4926, 48fnmpt2i 7519 . . . . . . . . . . . . . . 15 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏)
50 dffn4 6372 . . . . . . . . . . . . . . 15 ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) Fn (𝑎 × 𝑏) ↔ (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)))
5149, 50mpbi 222 . . . . . . . . . . . . . 14 (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))
52 fodomnum 9213 . . . . . . . . . . . . . 14 ((𝑎 × 𝑏) ∈ dom card → ((𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)):(𝑎 × 𝑏)–onto→ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏)))
5345, 51, 52mpisyl 21 . . . . . . . . . . . . 13 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏))
54 domtr 8294 . . . . . . . . . . . . 13 ((ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
5553, 43, 54syl2anc 579 . . . . . . . . . . . 12 ((𝑎 ≼ ω ∧ 𝑏 ≼ ω) → ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω)
5655ad2antrl 718 . . . . . . . . . . 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)
571, 2anim12i 606 . . . . . . . . . . . . . . 15 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ∈ Top ∧ 𝑆 ∈ Top))
5857ad3antrrr 720 . . . . . . . . . . . . . 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))
59 eltx 21780 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑧 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
6058, 59syl 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)))
61 eleq1 2847 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝑢, 𝑣⟩ → (𝑤 ∈ (𝑟 × 𝑠) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠)))
6261anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
63622rexbidv 3242 . . . . . . . . . . . . . . 15 (𝑤 = ⟨𝑢, 𝑣⟩ → (∃𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
6463rspccv 3508 . . . . . . . . . . . . . 14 (∀𝑤𝑧𝑟𝑅𝑠𝑆 (𝑤 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑟𝑅𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
65 r19.27v 3256 . . . . . . . . . . . . . . . . . 18 ((∀𝑟𝑅 (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∀𝑟𝑅 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))))
66 r19.29 3258 . . . . . . . . . . . . . . . . . . 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)))
67 r19.29 3258 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
68 opelxp 5391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ↔ (𝑢𝑟𝑣𝑠))
69 pm3.35 793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))
70 pm3.35 793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))
7169, 70anim12i 606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑢𝑟 ∧ (𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟))) ∧ (𝑣𝑠 ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7271an4s 650 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑢𝑟𝑣𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7368, 72sylanb 576 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) → (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
7473anim1i 608 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))) ∧ (𝑟 × 𝑠) ⊆ 𝑧) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7574anasss 460 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7675an12s 639 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
7776expl 451 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7877reximdv 3197 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) → (∃𝑠𝑆 ((𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧)))
7967, 78syl5 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)))
8079impl 449 . . . . . . . . . . . . . . . . . . . 20 ((((𝑢𝑟 → ∃𝑝𝑎 (𝑢𝑝𝑝𝑟)) ∧ ∀𝑠𝑆 (𝑣𝑠 → ∃𝑞𝑏 (𝑣𝑞𝑞𝑠))) ∧ ∃𝑠𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑧)) → ∃𝑠𝑆 ((∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)) ∧ (𝑟 × 𝑠) ⊆ 𝑧))
8180reximi 3192 . . . . . . . . . . . . . . . . . . 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))
8266, 81syl 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))
8365, 82sylan 575 . . . . . . . . . . . . . . . . 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))
84 reeanv 3293 . . . . . . . . . . . . . . . . . . . 20 (∃𝑝𝑎𝑞𝑏 ((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) ↔ (∃𝑝𝑎 (𝑢𝑝𝑝𝑟) ∧ ∃𝑞𝑏 (𝑣𝑞𝑞𝑠)))
85 simpr1l 1262 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
86 simpr1r 1264 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
87 eqidd 2779 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
88 xpeq1 5369 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑝 → (𝑚 × 𝑛) = (𝑝 × 𝑛))
8988eqeq2d 2788 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑝 → ((𝑝 × 𝑞) = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑛)))
90 xpeq2 5376 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑞 → (𝑝 × 𝑛) = (𝑝 × 𝑞))
9190eqeq2d 2788 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑞 → ((𝑝 × 𝑞) = (𝑝 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑝 × 𝑞)))
9289, 91rspc2ev 3526 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑎𝑞𝑏 ∧ (𝑝 × 𝑞) = (𝑝 × 𝑞)) → ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
9385, 86, 87, 92syl3anc 1439 . . . . . . . . . . . . . . . . . . . . . . . . 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))
94 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑝 ∈ V
95 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑞 ∈ V
9694, 95xpex 7240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 × 𝑞) ∈ V
97 eqeq1 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝑝 × 𝑞) → (𝑥 = (𝑚 × 𝑛) ↔ (𝑝 × 𝑞) = (𝑚 × 𝑛)))
98972rexbidv 3242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑝 × 𝑞) → (∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛) ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛)))
9996, 98elab 3558 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝 × 𝑞) ∈ {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)} ↔ ∃𝑚𝑎𝑛𝑏 (𝑝 × 𝑞) = (𝑚 × 𝑛))
10093, 99sylibr 226 . . . . . . . . . . . . . . . . . . . . . . . 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)})
10126rnmpt2 7047 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) = {𝑥 ∣ ∃𝑚𝑎𝑛𝑏 𝑥 = (𝑚 × 𝑛)}
102100, 101syl6eleqr 2870 . . . . . . . . . . . . . . . . . . . . . . 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)))
103 simpr2 1207 . . . . . . . . . . . . . . . . . . . . . . . 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)))
104 opelxpi 5392 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢𝑝𝑣𝑞) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
105104ad2ant2r 737 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞))
106103, 105syl 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))
107 xpss12 5370 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝𝑟𝑞𝑠) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
108107ad2ant2l 736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑢𝑝𝑝𝑟) ∧ (𝑣𝑞𝑞𝑠)) → (𝑝 × 𝑞) ⊆ (𝑟 × 𝑠))
109103, 108syl 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))
110 simpr3 1209 . . . . . . . . . . . . . . . . . . . . . . . 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)
111109, 110sstrd 3831 . . . . . . . . . . . . . . . . . . . . . . 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)
112 eleq2 2848 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (⟨𝑢, 𝑣⟩ ∈ 𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞)))
113 sseq1 3845 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑝 × 𝑞) → (𝑤𝑧 ↔ (𝑝 × 𝑞) ⊆ 𝑧))
114112, 113anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑝 × 𝑞) → ((⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)))
115114rspcev 3511 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝 × 𝑞) ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ∧ (⟨𝑢, 𝑣⟩ ∈ (𝑝 × 𝑞) ∧ (𝑝 × 𝑞) ⊆ 𝑧)) → ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))
116102, 106, 111, 115syl12anc 827 . . . . . . . . . . . . . . . . . . . . . 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))
1171163exp2 1416 . . . . . . . . . . . . . . . . . . . . 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)))))
118117rexlimdvv 3220 . . . . . . . . . . . . . . . . . . . 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))))
11984, 118syl5bir 235 . . . . . . . . . . . . . . . . . . 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))))
120119impd 400 . . . . . . . . . . . . . . . . . 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)))
121120rexlimdvva 3221 . . . . . . . . . . . . . . . . 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)))
12283, 121syl5 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)))
123122expd 406 . . . . . . . . . . . . . . 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))))
124123impr 448 . . . . . . . . . . . . . 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)))
12564, 124syl9r 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))))
12660, 125sylbid 232 . . . . . . . . . . . 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))))
127126ralrimiv 3147 . . . . . . . . . . 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)))
128 breq1 4889 . . . . . . . . . . . . 13 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (𝑦 ≼ ω ↔ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) ≼ ω))
129 rexeq 3331 . . . . . . . . . . . . . . 15 (𝑦 = ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛)) → (∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑚𝑎, 𝑛𝑏 ↦ (𝑚 × 𝑛))(⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
130129imbi2d 332 . . . . . . . . . . . . . 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))))
131130ralbidv 3168 . . . . . . . . . . . . 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))))
132128, 131anbi12d 624 . . . . . . . . . . . 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)))))
133132rspcev 3511 . . . . . . . . . . 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))))
13433, 56, 127, 133syl12anc 827 . . . . . . . . . 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))))
135134ex 403 . . . . . . . . 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)))))
13612, 135syl5bi 234 . . . . . . . 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)))))
137136rexlimdvva 3221 . . . . . . 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)))))
13811, 137syl5bir 235 . . . . . 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)))))
1397, 10, 138mp2and 689 . . . . 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))))
140139ralrimivva 3153 . . . 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))))
141 eleq1 2847 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑧 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑧))
142 eleq1 2847 . . . . . . . . . . 11 (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥𝑤 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑤))
143142anbi1d 623 . . . . . . . . . 10 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑤𝑤𝑧) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
144143rexbidv 3237 . . . . . . . . 9 (𝑥 = ⟨𝑢, 𝑣⟩ → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧)))
145141, 144imbi12d 336 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
146145ralbidv 3168 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → (∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝑅 ×t 𝑆)(⟨𝑢, 𝑣⟩ ∈ 𝑧 → ∃𝑤𝑦 (⟨𝑢, 𝑣⟩ ∈ 𝑤𝑤𝑧))))
147146anbi2d 622 . . . . . 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)))))
148147rexbidv 3237 . . . . 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)))))
149148ralxp 5509 . . . 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))))
150140, 149sylibr 226 . . 3 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ∀𝑥 ∈ ( 𝑅 × 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1515, 8txuni 21804 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1521, 2, 151syl2an 589 . . . 4 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
153152raleqdv 3340 . . 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)))))
154150, 153mpbid 224 . 2 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → ∀𝑥 (𝑅 ×t 𝑆)∃𝑦 ∈ 𝒫 (𝑅 ×t 𝑆)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝑅 ×t 𝑆)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
155 eqid 2778 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
156155is1stc2 21654 . 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)))))
1574, 154, 156sylanbrc 578 1 ((𝑅 ∈ 1st𝜔 ∧ 𝑆 ∈ 1st𝜔) → (𝑅 ×t 𝑆) ∈ 1st𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  {cab 2763  wral 3090  wrex 3091  wss 3792  𝒫 cpw 4379  cop 4404   cuni 4671   class class class wbr 4886   × cxp 5353  dom cdm 5355  ran crn 5356  Oncon0 5976   Fn wfn 6130  wf 6131  ontowfo 6133  (class class class)co 6922  cmpt2 6924  ωcom 7343  cen 8238  cdom 8239  cardccrd 9094  Topctop 21105  1st𝜔c1stc 21649   ×t ctx 21772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-oi 8704  df-card 9098  df-acn 9101  df-topgen 16490  df-top 21106  df-topon 21123  df-bases 21158  df-1stc 21651  df-tx 21774
This theorem is referenced by: (None)
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