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Theorem uptx 23633
Description: Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
uptx.1 𝑇 = (𝑅 ×t 𝑆)
uptx.2 𝑋 = 𝑅
uptx.3 𝑌 = 𝑆
uptx.4 𝑍 = (𝑋 × 𝑌)
uptx.5 𝑃 = (1st𝑍)
uptx.6 𝑄 = (2nd𝑍)
Assertion
Ref Expression
uptx ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Distinct variable groups:   ,𝐹   ,𝐺   𝑃,   𝑄,   𝑅,   𝑇,   𝑆,   𝑈,   ,𝑋   ,𝑌
Allowed substitution hint:   𝑍()

Proof of Theorem uptx
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 𝑈 = 𝑈
2 eqid 2737 . . . . 5 (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
31, 2txcnmpt 23632 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
4 uptx.1 . . . . 5 𝑇 = (𝑅 ×t 𝑆)
54oveq2i 7442 . . . 4 (𝑈 Cn 𝑇) = (𝑈 Cn (𝑅 ×t 𝑆))
63, 5eleqtrrdi 2852 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇))
7 uptx.2 . . . . . 6 𝑋 = 𝑅
81, 7cnf 23254 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹: 𝑈𝑋)
9 uptx.3 . . . . . 6 𝑌 = 𝑆
101, 9cnf 23254 . . . . 5 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺: 𝑈𝑌)
11 ffn 6736 . . . . . . . 8 (𝐹: 𝑈𝑋𝐹 Fn 𝑈)
1211adantr 480 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 Fn 𝑈)
13 fo1st 8034 . . . . . . . . . 10 1st :V–onto→V
14 fofn 6822 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1513, 14ax-mp 5 . . . . . . . . 9 1st Fn V
16 ssv 4008 . . . . . . . . 9 (𝑋 × 𝑌) ⊆ V
17 fnssres 6691 . . . . . . . . 9 ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
1815, 16, 17mp2an 692 . . . . . . . 8 (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
19 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑥 𝑈) → (𝐹𝑥) ∈ 𝑋)
20 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑥 𝑈) → (𝐺𝑥) ∈ 𝑌)
21 opelxpi 5722 . . . . . . . . . . . 12 (((𝐹𝑥) ∈ 𝑋 ∧ (𝐺𝑥) ∈ 𝑌) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2an 596 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑥 𝑈) ∧ (𝐺: 𝑈𝑌𝑥 𝑈)) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2322anandirs 679 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑥 𝑈) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2423fmpttd 7135 . . . . . . . . 9 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌))
25 ffn 6736 . . . . . . . . 9 ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝑈)
2624, 25syl 17 . . . . . . . 8 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝑈)
2724frnd 6744 . . . . . . . 8 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ran (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝑋 × 𝑌))
28 fnco 6686 . . . . . . . 8 (((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝑈 ∧ ran (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝑋 × 𝑌)) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
2918, 26, 27, 28mp3an2i 1468 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
30 fvco3 7008 . . . . . . . . 9 (((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌) ∧ 𝑧 𝑈) → (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
3124, 30sylan 580 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
32 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
33 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
3432, 33opeq12d 4881 . . . . . . . . . . 11 (𝑥 = 𝑧 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
35 opex 5469 . . . . . . . . . . 11 ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ V
3634, 2, 35fvmpt 7016 . . . . . . . . . 10 (𝑧 𝑈 → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
3736adantl 481 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
3837fveq2d 6910 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
39 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑧 𝑈) → (𝐹𝑧) ∈ 𝑋)
40 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑧 𝑈) → (𝐺𝑧) ∈ 𝑌)
41 opelxpi 5722 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ 𝑋 ∧ (𝐺𝑧) ∈ 𝑌) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4239, 40, 41syl2an 596 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑧 𝑈) ∧ (𝐺: 𝑈𝑌𝑧 𝑈)) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4342anandirs 679 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4443fvresd 6926 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
45 fvex 6919 . . . . . . . . . 10 (𝐹𝑧) ∈ V
46 fvex 6919 . . . . . . . . . 10 (𝐺𝑧) ∈ V
4745, 46op1st 8022 . . . . . . . . 9 (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧)
4844, 47eqtrdi 2793 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
4931, 38, 483eqtrrd 2782 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐹𝑧) = (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
5012, 29, 49eqfnfvd 7054 . . . . . 6 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
51 uptx.5 . . . . . . . 8 𝑃 = (1st𝑍)
52 uptx.4 . . . . . . . . 9 𝑍 = (𝑋 × 𝑌)
5352reseq2i 5994 . . . . . . . 8 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
5451, 53eqtri 2765 . . . . . . 7 𝑃 = (1st ↾ (𝑋 × 𝑌))
5554coeq1i 5870 . . . . . 6 (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
5650, 55eqtr4di 2795 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
578, 10, 56syl2an 596 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
58 ffn 6736 . . . . . . . 8 (𝐺: 𝑈𝑌𝐺 Fn 𝑈)
5958adantl 481 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 Fn 𝑈)
60 fo2nd 8035 . . . . . . . . . 10 2nd :V–onto→V
61 fofn 6822 . . . . . . . . . 10 (2nd :V–onto→V → 2nd Fn V)
6260, 61ax-mp 5 . . . . . . . . 9 2nd Fn V
63 fnssres 6691 . . . . . . . . 9 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6462, 16, 63mp2an 692 . . . . . . . 8 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
65 fnco 6686 . . . . . . . 8 (((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝑈 ∧ ran (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝑋 × 𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
6664, 26, 27, 65mp3an2i 1468 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
67 fvco3 7008 . . . . . . . . 9 (((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌) ∧ 𝑧 𝑈) → (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
6824, 67sylan 580 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
6937fveq2d 6910 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
7043fvresd 6926 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
7145, 46op2nd 8023 . . . . . . . . 9 (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧)
7270, 71eqtrdi 2793 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
7368, 69, 723eqtrrd 2782 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐺𝑧) = (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
7459, 66, 73eqfnfvd 7054 . . . . . 6 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
75 uptx.6 . . . . . . . 8 𝑄 = (2nd𝑍)
7652reseq2i 5994 . . . . . . . 8 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
7775, 76eqtri 2765 . . . . . . 7 𝑄 = (2nd ↾ (𝑋 × 𝑌))
7877coeq1i 5870 . . . . . 6 (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
7974, 78eqtr4di 2795 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
808, 10, 79syl2an 596 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
816, 57, 80jca32 515 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))))
82 eleq1 2829 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ( ∈ (𝑈 Cn 𝑇) ↔ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇)))
83 coeq2 5869 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑃) = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8483eqeq2d 2748 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐹 = (𝑃) ↔ 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
85 coeq2 5869 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑄) = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8685eqeq2d 2748 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐺 = (𝑄) ↔ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
8784, 86anbi12d 632 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ((𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))))
8882, 87anbi12d 632 . . . 4 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ↔ ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))))
8988spcegv 3597 . . 3 ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇) → (((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))) → ∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
906, 81, 89sylc 65 . 2 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
91 eqid 2737 . . . . . . . 8 𝑇 = 𝑇
921, 91cnf 23254 . . . . . . 7 ( ∈ (𝑈 Cn 𝑇) → : 𝑈 𝑇)
93 cntop2 23249 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
94 cntop2 23249 . . . . . . . . 9 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
954unieqi 4919 . . . . . . . . . 10 𝑇 = (𝑅 ×t 𝑆)
967, 9txuni 23600 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
9795, 96eqtr4id 2796 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑇 = (𝑋 × 𝑌))
9893, 94, 97syl2an 596 . . . . . . . 8 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑇 = (𝑋 × 𝑌))
9998feq3d 6723 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (: 𝑈 𝑇: 𝑈⟶(𝑋 × 𝑌)))
10092, 99imbitrid 244 . . . . . 6 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ( ∈ (𝑈 Cn 𝑇) → : 𝑈⟶(𝑋 × 𝑌)))
101100anim1d 611 . . . . 5 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
102 3anass 1095 . . . . 5 ((: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (: 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
103101, 102imbitrrdi 252 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
104103alrimiv 1927 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀(( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
105 cntop1 23248 . . . . . 6 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
106105uniexd 7762 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ V)
10754, 77upxp 23631 . . . . 5 (( 𝑈 ∈ V ∧ 𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
108106, 8, 10, 107syl2an3an 1424 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
109 eumo 2578 . . . 4 (∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
110108, 109syl 17 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
111 moim 2544 . . 3 (∀(( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
112104, 110, 111sylc 65 . 2 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
113 df-reu 3381 . . 3 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
114 df-eu 2569 . . 3 (∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
115113, 114bitri 275 . 2 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
11690, 112, 115sylanbrc 583 1 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  ∃!wreu 3378  Vcvv 3480  wss 3951  cop 4632   cuni 4907  cmpt 5225   × cxp 5683  ran crn 5686  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Topctop 22899   Cn ccn 23232   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-tx 23570
This theorem is referenced by:  txcn  23634
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