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Theorem uptx 22149
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 2824 . . . . 5 𝑈 = 𝑈
2 eqid 2824 . . . . 5 (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
31, 2txcnmpt 22148 . . . 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 7162 . . . 4 (𝑈 Cn 𝑇) = (𝑈 Cn (𝑅 ×t 𝑆))
63, 5syl6eleqr 2928 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇))
7 uptx.2 . . . . . 6 𝑋 = 𝑅
81, 7cnf 21770 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹: 𝑈𝑋)
9 uptx.3 . . . . . 6 𝑌 = 𝑆
101, 9cnf 21770 . . . . 5 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺: 𝑈𝑌)
11 ffn 6510 . . . . . . . 8 (𝐹: 𝑈𝑋𝐹 Fn 𝑈)
1211adantr 481 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 Fn 𝑈)
13 fo1st 7703 . . . . . . . . . 10 1st :V–onto→V
14 fofn 6588 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1513, 14ax-mp 5 . . . . . . . . 9 1st Fn V
16 ssv 3994 . . . . . . . . 9 (𝑋 × 𝑌) ⊆ V
17 fnssres 6466 . . . . . . . . 9 ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
1815, 16, 17mp2an 688 . . . . . . . 8 (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
19 ffvelrn 6844 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑥 𝑈) → (𝐹𝑥) ∈ 𝑋)
20 ffvelrn 6844 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑥 𝑈) → (𝐺𝑥) ∈ 𝑌)
21 opelxpi 5590 . . . . . . . . . . . 12 (((𝐹𝑥) ∈ 𝑋 ∧ (𝐺𝑥) ∈ 𝑌) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2an 595 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑥 𝑈) ∧ (𝐺: 𝑈𝑌𝑥 𝑈)) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2322anandirs 675 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑥 𝑈) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2423fmpttd 6874 . . . . . . . . 9 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌))
25 ffn 6510 . . . . . . . . 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 6517 . . . . . . . 8 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ran (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝑋 × 𝑌))
28 fnco 6461 . . . . . . . 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 1459 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
30 fvco3 6756 . . . . . . . . 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 6666 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
33 fveq2 6666 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
3432, 33opeq12d 4809 . . . . . . . . . . 11 (𝑥 = 𝑧 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
35 opex 5352 . . . . . . . . . . 11 ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ V
3634, 2, 35fvmpt 6764 . . . . . . . . . 10 (𝑧 𝑈 → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
3736adantl 482 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
3837fveq2d 6670 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
39 ffvelrn 6844 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑧 𝑈) → (𝐹𝑧) ∈ 𝑋)
40 ffvelrn 6844 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑧 𝑈) → (𝐺𝑧) ∈ 𝑌)
41 opelxpi 5590 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ 𝑋 ∧ (𝐺𝑧) ∈ 𝑌) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4239, 40, 41syl2an 595 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑧 𝑈) ∧ (𝐺: 𝑈𝑌𝑧 𝑈)) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4342anandirs 675 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4443fvresd 6686 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
45 fvex 6679 . . . . . . . . . 10 (𝐹𝑧) ∈ V
46 fvex 6679 . . . . . . . . . 10 (𝐺𝑧) ∈ V
4745, 46op1st 7691 . . . . . . . . 9 (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧)
4844, 47syl6eq 2876 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
4931, 38, 483eqtrrd 2865 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐹𝑧) = (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
5012, 29, 49eqfnfvd 6800 . . . . . 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 5848 . . . . . . . 8 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
5451, 53eqtri 2848 . . . . . . 7 𝑃 = (1st ↾ (𝑋 × 𝑌))
5554coeq1i 5728 . . . . . 6 (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
5650, 55syl6eqr 2878 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
578, 10, 56syl2an 595 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
58 ffn 6510 . . . . . . . 8 (𝐺: 𝑈𝑌𝐺 Fn 𝑈)
5958adantl 482 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 Fn 𝑈)
60 fo2nd 7704 . . . . . . . . . 10 2nd :V–onto→V
61 fofn 6588 . . . . . . . . . 10 (2nd :V–onto→V → 2nd Fn V)
6260, 61ax-mp 5 . . . . . . . . 9 2nd Fn V
63 fnssres 6466 . . . . . . . . 9 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6462, 16, 63mp2an 688 . . . . . . . 8 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
65 fnco 6461 . . . . . . . 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 1459 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
67 fvco3 6756 . . . . . . . . 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 6670 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
7043fvresd 6686 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
7145, 46op2nd 7692 . . . . . . . . 9 (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧)
7270, 71syl6eq 2876 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
7368, 69, 723eqtrrd 2865 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐺𝑧) = (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
7459, 66, 73eqfnfvd 6800 . . . . . 6 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
75 uptx.6 . . . . . . . 8 𝑄 = (2nd𝑍)
7652reseq2i 5848 . . . . . . . 8 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
7775, 76eqtri 2848 . . . . . . 7 𝑄 = (2nd ↾ (𝑋 × 𝑌))
7877coeq1i 5728 . . . . . 6 (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
7974, 78syl6eqr 2878 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
808, 10, 79syl2an 595 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
816, 57, 80jca32 516 . . 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 2904 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ( ∈ (𝑈 Cn 𝑇) ↔ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇)))
83 coeq2 5727 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑃) = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8483eqeq2d 2835 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐹 = (𝑃) ↔ 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
85 coeq2 5727 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑄) = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8685eqeq2d 2835 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐺 = (𝑄) ↔ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
8784, 86anbi12d 630 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ((𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))))
8882, 87anbi12d 630 . . . 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 3601 . . 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 2824 . . . . . . . 8 𝑇 = 𝑇
921, 91cnf 21770 . . . . . . 7 ( ∈ (𝑈 Cn 𝑇) → : 𝑈 𝑇)
93 cntop2 21765 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
94 cntop2 21765 . . . . . . . . 9 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
957, 9txuni 22116 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
964unieqi 4845 . . . . . . . . . 10 𝑇 = (𝑅 ×t 𝑆)
9795, 96syl6reqr 2879 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑇 = (𝑋 × 𝑌))
9893, 94, 97syl2an 595 . . . . . . . 8 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑇 = (𝑋 × 𝑌))
9998feq3d 6497 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (: 𝑈 𝑇: 𝑈⟶(𝑋 × 𝑌)))
10092, 99syl5ib 245 . . . . . 6 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ( ∈ (𝑈 Cn 𝑇) → : 𝑈⟶(𝑋 × 𝑌)))
101100anim1d 610 . . . . 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 1089 . . . . 5 ((: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (: 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
103101, 102syl6ibr 253 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
104103alrimiv 1921 . . 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 21764 . . . . . 6 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
106 uniexg 7460 . . . . . 6 (𝑈 ∈ Top → 𝑈 ∈ V)
107105, 106syl 17 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ V)
10854, 77upxp 22147 . . . . 5 (( 𝑈 ∈ V ∧ 𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
109107, 8, 10, 108syl2an3an 1416 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
110 eumo 2658 . . . 4 (∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
111109, 110syl 17 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
112 moim 2619 . . 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)))))
113104, 111, 112sylc 65 . 2 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
114 df-reu 3149 . . 3 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
115 df-eu 2649 . . 3 (∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
116114, 115bitri 276 . 2 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
11790, 113, 116sylanbrc 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 396  w3a 1081  wal 1528   = wceq 1530  wex 1773  wcel 2106  ∃*wmo 2613  ∃!weu 2648  ∃!wreu 3144  Vcvv 3499  wss 3939  cop 4569   cuni 4836  cmpt 5142   × cxp 5551  ran crn 5554  cres 5555  ccom 5557   Fn wfn 6346  wf 6347  ontowfo 6349  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  Topctop 21417   Cn ccn 21748   ×t ctx 22084
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-topgen 16709  df-top 21418  df-topon 21435  df-bases 21470  df-cn 21751  df-tx 22086
This theorem is referenced by:  txcn  22150
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