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Theorem uptx 13407
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 2177 . . . . 5 𝑈 = 𝑈
2 eqid 2177 . . . . 5 (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
31, 2txcnmpt 13406 . . . 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 5879 . . . 4 (𝑈 Cn 𝑇) = (𝑈 Cn (𝑅 ×t 𝑆))
63, 5eleqtrrdi 2271 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇))
7 uptx.2 . . . . . 6 𝑋 = 𝑅
81, 7cnf 13337 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹: 𝑈𝑋)
9 uptx.3 . . . . . 6 𝑌 = 𝑆
101, 9cnf 13337 . . . . 5 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺: 𝑈𝑌)
11 ffn 5360 . . . . . . . 8 (𝐹: 𝑈𝑋𝐹 Fn 𝑈)
1211adantr 276 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 Fn 𝑈)
13 fo1st 6151 . . . . . . . . . 10 1st :V–onto→V
14 fofn 5435 . . . . . . . . . 10 (1st :V–onto→V → 1st Fn V)
1513, 14ax-mp 5 . . . . . . . . 9 1st Fn V
16 ssv 3177 . . . . . . . . 9 (𝑋 × 𝑌) ⊆ V
17 fnssres 5324 . . . . . . . . 9 ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
1815, 16, 17mp2an 426 . . . . . . . 8 (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
19 ffvelcdm 5644 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑥 𝑈) → (𝐹𝑥) ∈ 𝑋)
20 ffvelcdm 5644 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑥 𝑈) → (𝐺𝑥) ∈ 𝑌)
21 opelxpi 4654 . . . . . . . . . . . 12 (((𝐹𝑥) ∈ 𝑋 ∧ (𝐺𝑥) ∈ 𝑌) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2an 289 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑥 𝑈) ∧ (𝐺: 𝑈𝑌𝑥 𝑈)) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2322anandirs 593 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑥 𝑈) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝑋 × 𝑌))
2423fmpttd 5666 . . . . . . . . 9 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩): 𝑈⟶(𝑋 × 𝑌))
2524ffnd 5361 . . . . . . . 8 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝑈)
2624frnd 5370 . . . . . . . 8 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ran (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝑋 × 𝑌))
27 fnco 5319 . . . . . . . 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)
2818, 25, 26, 27mp3an2i 1342 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
29 fvco3 5582 . . . . . . . . 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)))
3024, 29sylan 283 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
31 fveq2 5510 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
32 fveq2 5510 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
3331, 32opeq12d 3784 . . . . . . . . . 10 (𝑥 = 𝑧 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
34 simpr 110 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → 𝑧 𝑈)
35 simpll 527 . . . . . . . . . . . 12 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → 𝐹: 𝑈𝑋)
3635, 34ffvelcdmd 5647 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐹𝑧) ∈ 𝑋)
37 simplr 528 . . . . . . . . . . . 12 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → 𝐺: 𝑈𝑌)
3837, 34ffvelcdmd 5647 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐺𝑧) ∈ 𝑌)
3936, 38opelxpd 4655 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
402, 33, 34, 39fvmptd3 5604 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
4140fveq2d 5514 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
42 ffvelcdm 5644 . . . . . . . . . . . 12 ((𝐹: 𝑈𝑋𝑧 𝑈) → (𝐹𝑧) ∈ 𝑋)
43 ffvelcdm 5644 . . . . . . . . . . . 12 ((𝐺: 𝑈𝑌𝑧 𝑈) → (𝐺𝑧) ∈ 𝑌)
44 opelxpi 4654 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ 𝑋 ∧ (𝐺𝑧) ∈ 𝑌) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4542, 43, 44syl2an 289 . . . . . . . . . . 11 (((𝐹: 𝑈𝑋𝑧 𝑈) ∧ (𝐺: 𝑈𝑌𝑧 𝑈)) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4645anandirs 593 . . . . . . . . . 10 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝑋 × 𝑌))
4746fvresd 5535 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
48 op1stg 6144 . . . . . . . . . 10 (((𝐹𝑧) ∈ 𝑋 ∧ (𝐺𝑧) ∈ 𝑌) → (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
4936, 38, 48syl2anc 411 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
5047, 49eqtrd 2210 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
5130, 41, 503eqtrrd 2215 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐹𝑧) = (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
5212, 28, 51eqfnfvd 5611 . . . . . 6 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
53 uptx.5 . . . . . . . 8 𝑃 = (1st𝑍)
54 uptx.4 . . . . . . . . 9 𝑍 = (𝑋 × 𝑌)
5554reseq2i 4899 . . . . . . . 8 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
5653, 55eqtri 2198 . . . . . . 7 𝑃 = (1st ↾ (𝑋 × 𝑌))
5756coeq1i 4781 . . . . . 6 (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
5852, 57eqtr4di 2228 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
598, 10, 58syl2an 289 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
60 ffn 5360 . . . . . . . 8 (𝐺: 𝑈𝑌𝐺 Fn 𝑈)
6160adantl 277 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 Fn 𝑈)
62 fo2nd 6152 . . . . . . . . . 10 2nd :V–onto→V
63 fofn 5435 . . . . . . . . . 10 (2nd :V–onto→V → 2nd Fn V)
6462, 63ax-mp 5 . . . . . . . . 9 2nd Fn V
65 fnssres 5324 . . . . . . . . 9 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6664, 16, 65mp2an 426 . . . . . . . 8 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
67 fnco 5319 . . . . . . . 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)
6866, 25, 26, 67mp3an2i 1342 . . . . . . 7 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝑈)
69 fvco3 5582 . . . . . . . . 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)))
7024, 69sylan 283 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
7140fveq2d 5514 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
7246fvresd 5535 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
73 op2ndg 6145 . . . . . . . . . 10 (((𝐹𝑧) ∈ 𝑋 ∧ (𝐺𝑧) ∈ 𝑌) → (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
7436, 38, 73syl2anc 411 . . . . . . . . 9 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
7572, 74eqtrd 2210 . . . . . . . 8 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
7670, 71, 753eqtrrd 2215 . . . . . . 7 (((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) ∧ 𝑧 𝑈) → (𝐺𝑧) = (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
7761, 68, 76eqfnfvd 5611 . . . . . 6 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
78 uptx.6 . . . . . . . 8 𝑄 = (2nd𝑍)
7954reseq2i 4899 . . . . . . . 8 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
8078, 79eqtri 2198 . . . . . . 7 𝑄 = (2nd ↾ (𝑋 × 𝑌))
8180coeq1i 4781 . . . . . 6 (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
8277, 81eqtr4di 2228 . . . . 5 ((𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
838, 10, 82syl2an 289 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
846, 59, 83jca32 310 . . 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)⟩)))))
85 eleq1 2240 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ( ∈ (𝑈 Cn 𝑇) ↔ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ (𝑈 Cn 𝑇)))
86 coeq2 4780 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑃) = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8786eqeq2d 2189 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐹 = (𝑃) ↔ 𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
88 coeq2 4780 . . . . . . 7 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑄) = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
8988eqeq2d 2189 . . . . . 6 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐺 = (𝑄) ↔ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
9087, 89anbi12d 473 . . . . 5 ( = (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ((𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (𝐹 = (𝑃 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 𝑈 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))))
9185, 90anbi12d 473 . . . 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)⟩))))))
9291spcegv 2825 . . 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)))))
936, 84, 92sylc 62 . 2 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
94 eqid 2177 . . . . . . . 8 𝑇 = 𝑇
951, 94cnf 13337 . . . . . . 7 ( ∈ (𝑈 Cn 𝑇) → : 𝑈 𝑇)
96 cntop2 13335 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
97 cntop2 13335 . . . . . . . . 9 (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
984unieqi 3817 . . . . . . . . . 10 𝑇 = (𝑅 ×t 𝑆)
997, 9txuni 13396 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
10098, 99eqtr4id 2229 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑇 = (𝑋 × 𝑌))
10196, 97, 100syl2an 289 . . . . . . . 8 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑇 = (𝑋 × 𝑌))
102101feq3d 5349 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (: 𝑈 𝑇: 𝑈⟶(𝑋 × 𝑌)))
10395, 102syl5ib 154 . . . . . 6 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ( ∈ (𝑈 Cn 𝑇) → : 𝑈⟶(𝑋 × 𝑌)))
104103anim1d 336 . . . . 5 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
105 3anass 982 . . . . 5 ((: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (: 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
106104, 105syl6ibr 162 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
107106alrimiv 1874 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀(( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
108 cntop1 13334 . . . . . 6 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
109 uniexg 4435 . . . . . 6 (𝑈 ∈ Top → 𝑈 ∈ V)
110108, 109syl 14 . . . . 5 (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ V)
11156, 80upxp 13405 . . . . 5 (( 𝑈 ∈ V ∧ 𝐹: 𝑈𝑋𝐺: 𝑈𝑌) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
112110, 8, 10, 111syl2an3an 1298 . . . 4 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
113 eumo 2058 . . . 4 (∃!(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
114112, 113syl 14 . . 3 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*(: 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
115 moim 2090 . . 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)))))
116107, 114, 115sylc 62 . 2 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
117 df-reu 2462 . . 3 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
118 eu5 2073 . . 3 (∃!( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
119117, 118bitri 184 . 2 (∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ (∃( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ ∃*( ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))))
12093, 116, 119sylanbrc 417 1 ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wal 1351   = wceq 1353  wex 1492  ∃!weu 2026  ∃*wmo 2027  wcel 2148  ∃!wreu 2457  Vcvv 2737  wss 3129  cop 3594   cuni 3807  cmpt 4061   × cxp 4620  ran crn 4623  cres 4624  ccom 4626   Fn wfn 5206  wf 5207  ontowfo 5209  cfv 5211  (class class class)co 5868  1st c1st 6132  2nd c2nd 6133  Topctop 13128   Cn ccn 13318   ×t ctx 13385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-topgen 12644  df-top 13129  df-topon 13142  df-bases 13174  df-cn 13321  df-tx 13386
This theorem is referenced by:  txcn  13408
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