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Theorem ptunhmeo 21421
 Description: Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of (𝐴↑𝐵) · (𝐴↑𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
ptunhmeo.x 𝑋 = 𝐾
ptunhmeo.y 𝑌 = 𝐿
ptunhmeo.j 𝐽 = (∏t𝐹)
ptunhmeo.k 𝐾 = (∏t‘(𝐹𝐴))
ptunhmeo.l 𝐿 = (∏t‘(𝐹𝐵))
ptunhmeo.g 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
ptunhmeo.c (𝜑𝐶𝑉)
ptunhmeo.f (𝜑𝐹:𝐶⟶Top)
ptunhmeo.u (𝜑𝐶 = (𝐴𝐵))
ptunhmeo.i (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
ptunhmeo (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ptunhmeo
Dummy variables 𝑓 𝑘 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptunhmeo.g . . . . 5 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
2 vex 3176 . . . . . . . 8 𝑥 ∈ V
3 vex 3176 . . . . . . . 8 𝑦 ∈ V
42, 3op1std 7069 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
52, 3op2ndd 7070 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
64, 5uneq12d 3730 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
76mpt2mpt 6650 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2635 . . . 4 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧)))
9 xp1st 7089 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
109adantl 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ 𝑋)
11 ixpeq2 7808 . . . . . . . . . . . . 13 (∀𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛))
12 fvres 6117 . . . . . . . . . . . . . 14 (𝑛𝐴 → ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1312unieqd 4382 . . . . . . . . . . . . 13 (𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1411, 13mprg 2910 . . . . . . . . . . . 12 X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛)
15 ptunhmeo.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑉)
16 ssun1 3738 . . . . . . . . . . . . . . 15 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . . . . 15 (𝜑𝐶 = (𝐴𝐵))
1816, 17syl5sseqr 3617 . . . . . . . . . . . . . 14 (𝜑𝐴𝐶)
1915, 18ssexd 4733 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 5984 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . . . . 14 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 21207 . . . . . . . . . . . . 13 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2419, 21, 23syl2anc 691 . . . . . . . . . . . 12 (𝜑X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2514, 24syl5eqr 2658 . . . . . . . . . . 11 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝐾)
26 ptunhmeo.x . . . . . . . . . . 11 𝑋 = 𝐾
2725, 26syl6eqr 2662 . . . . . . . . . 10 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝑋)
2827adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
2910, 28eleqtrrd 2691 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛))
30 xp2nd 7090 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
3130adantl 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ 𝑌)
3217eqcomd 2616 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . . . . 14 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4009 . . . . . . . . . . . . . 14 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 691 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 221 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 7806 . . . . . . . . . . 11 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = X𝑛𝐵 (𝐹𝑛))
38 ixpeq2 7808 . . . . . . . . . . . . . 14 (∀𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛))
39 fvres 6117 . . . . . . . . . . . . . . 15 (𝑛𝐵 → ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4039unieqd 4382 . . . . . . . . . . . . . 14 (𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4138, 40mprg 2910 . . . . . . . . . . . . 13 X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛)
42 ssun2 3739 . . . . . . . . . . . . . . . 16 𝐵 ⊆ (𝐴𝐵)
4342, 17syl5sseqr 3617 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
4415, 43ssexd 4733 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
4520, 43fssresd 5984 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . . . . 15 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 21207 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4844, 45, 47syl2anc 691 . . . . . . . . . . . . 13 (𝜑X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4941, 48syl5eqr 2658 . . . . . . . . . . . 12 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝐿)
50 ptunhmeo.y . . . . . . . . . . . 12 𝑌 = 𝐿
5149, 50syl6eqr 2662 . . . . . . . . . . 11 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝑌)
5237, 51eqtrd 2644 . . . . . . . . . 10 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5352adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5431, 53eleqtrrd 2691 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛))
5518adantr 480 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 7830 . . . . . . . 8 (((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) ∧ (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) ∧ 𝐴𝐶) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
5729, 54, 55, 56syl3anc 1318 . . . . . . 7 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
58 ixpfn 7800 . . . . . . 7 (((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
5957, 58syl 17 . . . . . 6 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
60 dffn5 6151 . . . . . 6 (((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶 ↔ ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6159, 60sylib 207 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6261mpteq2dva 4672 . . . 4 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
638, 62syl5eq 2656 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
64 ptunhmeo.j . . . 4 𝐽 = (∏t𝐹)
65 pttop 21195 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → (∏t‘(𝐹𝐴)) ∈ Top)
6619, 21, 65syl2anc 691 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐴)) ∈ Top)
6722, 66syl5eqel 2692 . . . . . 6 (𝜑𝐾 ∈ Top)
6826toptopon 20548 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑋))
6967, 68sylib 207 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑋))
70 pttop 21195 . . . . . . . 8 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → (∏t‘(𝐹𝐵)) ∈ Top)
7144, 45, 70syl2anc 691 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐵)) ∈ Top)
7246, 71syl5eqel 2692 . . . . . 6 (𝜑𝐿 ∈ Top)
7350toptopon 20548 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘𝑌))
7472, 73sylib 207 . . . . 5 (𝜑𝐿 ∈ (TopOn‘𝑌))
75 txtopon 21204 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7669, 74, 75syl2anc 691 . . . 4 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7717eleq2d 2673 . . . . . . 7 (𝜑 → (𝑘𝐶𝑘 ∈ (𝐴𝐵)))
7877biimpa 500 . . . . . 6 ((𝜑𝑘𝐶) → 𝑘 ∈ (𝐴𝐵))
79 elun 3715 . . . . . 6 (𝑘 ∈ (𝐴𝐵) ↔ (𝑘𝐴𝑘𝐵))
8078, 79sylib 207 . . . . 5 ((𝜑𝑘𝐶) → (𝑘𝐴𝑘𝐵))
81 ixpfn 7800 . . . . . . . . . . 11 ((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) → (1st𝑧) Fn 𝐴)
8229, 81syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8382adantlr 747 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8451adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐵 (𝐹𝑛) = 𝑌)
8531, 84eleqtrrd 2691 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛))
86 ixpfn 7800 . . . . . . . . . . 11 ((2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛) → (2nd𝑧) Fn 𝐵)
8785, 86syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8887adantlr 747 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8933ad2antrr 758 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
90 simplr 788 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐴)
91 fvun1 6179 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐴)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9283, 88, 89, 90, 91syl112anc 1322 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9392mpteq2dva 4672 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)))
9476adantr 480 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
954mpt2mpt 6650 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
9669adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐾 ∈ (TopOn‘𝑋))
9774adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐿 ∈ (TopOn‘𝑌))
9896, 97cnmpt1st 21281 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
9995, 98syl5eqel 2692 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
10019adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐴 ∈ V)
10121adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝐴):𝐴⟶Top)
102 simpr 476 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝑘𝐴)
10326, 22ptpjcn 21224 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top ∧ 𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
104100, 101, 102, 103syl3anc 1318 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
105 fvres 6117 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
106105adantl 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
107106oveq2d 6565 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝐾 Cn ((𝐹𝐴)‘𝑘)) = (𝐾 Cn (𝐹𝑘)))
108104, 107eleqtrd 2690 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn (𝐹𝑘)))
109 fveq1 6102 . . . . . . . 8 (𝑓 = (1st𝑧) → (𝑓𝑘) = ((1st𝑧)‘𝑘))
11094, 99, 96, 108, 109cnmpt11 21276 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11193, 110eqeltrd 2688 . . . . . 6 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11282adantlr 747 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
11387adantlr 747 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
11433ad2antrr 758 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
115 simplr 788 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐵)
116 fvun2 6180 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐵)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
117112, 113, 114, 115, 116syl112anc 1322 . . . . . . . 8 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
118117mpteq2dva 4672 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)))
11976adantr 480 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
1205mpt2mpt 6650 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
12169adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐾 ∈ (TopOn‘𝑋))
12274adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐿 ∈ (TopOn‘𝑌))
123121, 122cnmpt2nd 21282 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
124120, 123syl5eqel 2692 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
12544adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐵 ∈ V)
12645adantr 480 . . . . . . . . . 10 ((𝜑𝑘𝐵) → (𝐹𝐵):𝐵⟶Top)
127 simpr 476 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝑘𝐵)
12850, 46ptpjcn 21224 . . . . . . . . . 10 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top ∧ 𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
129125, 126, 127, 128syl3anc 1318 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
130 fvres 6117 . . . . . . . . . . 11 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
131130adantl 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
132131oveq2d 6565 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝐿 Cn ((𝐹𝐵)‘𝑘)) = (𝐿 Cn (𝐹𝑘)))
133129, 132eleqtrd 2690 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn (𝐹𝑘)))
134 fveq1 6102 . . . . . . . 8 (𝑓 = (2nd𝑧) → (𝑓𝑘) = ((2nd𝑧)‘𝑘))
135119, 124, 122, 133, 134cnmpt11 21276 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
136118, 135eqeltrd 2688 . . . . . 6 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
137111, 136jaodan 822 . . . . 5 ((𝜑 ∧ (𝑘𝐴𝑘𝐵)) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13880, 137syldan 486 . . . 4 ((𝜑𝑘𝐶) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13964, 76, 15, 20, 138ptcn 21240 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14063, 139eqeltrd 2688 . 2 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14126, 50, 64, 22, 46, 1, 15, 20, 17, 33ptuncnv 21420 . . 3 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
142 pttop 21195 . . . . . . 7 ((𝐶𝑉𝐹:𝐶⟶Top) → (∏t𝐹) ∈ Top)
14315, 20, 142syl2anc 691 . . . . . 6 (𝜑 → (∏t𝐹) ∈ Top)
14464, 143syl5eqel 2692 . . . . 5 (𝜑𝐽 ∈ Top)
145 eqid 2610 . . . . . 6 𝐽 = 𝐽
146145toptopon 20548 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
147144, 146sylib 207 . . . 4 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
148145, 64, 22ptrescn 21252 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐴𝐶) → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
14915, 20, 18, 148syl3anc 1318 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
150145, 64, 46ptrescn 21252 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐵𝐶) → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
15115, 20, 43, 150syl3anc 1318 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
152147, 149, 151cnmpt1t 21278 . . 3 (𝜑 → (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
153141, 152eqeltrd 2688 . 2 (𝜑𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
154 ishmeo 21372 . 2 (𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽) ↔ (𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽) ∧ 𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿))))
155140, 153, 154sylanbrc 695 1 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Xcixp 7794  ∏tcpt 15922  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ×t ctx 21173  Homeochmeo 21366 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 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-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-iin 4458  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-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-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-ixp 7795  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368 This theorem is referenced by:  xpstopnlem1  21422  ptcmpfi  21426
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