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Theorem ptunhmeo 22705
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 (𝐴𝐵) · (𝐴𝐶) = 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 3412 . . . . . . . 8 𝑥 ∈ V
3 vex 3412 . . . . . . . 8 𝑦 ∈ V
42, 3op1std 7771 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
52, 3op2ndd 7772 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
64, 5uneq12d 4078 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
76mpompt 7324 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2768 . . . 4 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧)))
9 xp1st 7793 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
109adantl 485 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ 𝑋)
11 ixpeq2 8592 . . . . . . . . . . . . 13 (∀𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛))
12 fvres 6736 . . . . . . . . . . . . . 14 (𝑛𝐴 → ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1312unieqd 4833 . . . . . . . . . . . . 13 (𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1411, 13mprg 3075 . . . . . . . . . . . 12 X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛)
15 ptunhmeo.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑉)
16 ssun1 4086 . . . . . . . . . . . . . . 15 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . . . . 15 (𝜑𝐶 = (𝐴𝐵))
1816, 17sseqtrrid 3954 . . . . . . . . . . . . . 14 (𝜑𝐴𝐶)
1915, 18ssexd 5217 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6586 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . . . . 14 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 22491 . . . . . . . . . . . . 13 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2419, 21, 23syl2anc 587 . . . . . . . . . . . 12 (𝜑X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2514, 24eqtr3id 2792 . . . . . . . . . . 11 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝐾)
26 ptunhmeo.x . . . . . . . . . . 11 𝑋 = 𝐾
2725, 26eqtr4di 2796 . . . . . . . . . 10 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝑋)
2827adantr 484 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
2910, 28eleqtrrd 2841 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛))
30 xp2nd 7794 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
3130adantl 485 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ 𝑌)
3217eqcomd 2743 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . . . . 14 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4404 . . . . . . . . . . . . . 14 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 235 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 8590 . . . . . . . . . . 11 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = X𝑛𝐵 (𝐹𝑛))
38 ixpeq2 8592 . . . . . . . . . . . . . 14 (∀𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛))
39 fvres 6736 . . . . . . . . . . . . . . 15 (𝑛𝐵 → ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4039unieqd 4833 . . . . . . . . . . . . . 14 (𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4138, 40mprg 3075 . . . . . . . . . . . . 13 X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛)
42 ssun2 4087 . . . . . . . . . . . . . . . 16 𝐵 ⊆ (𝐴𝐵)
4342, 17sseqtrrid 3954 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
4415, 43ssexd 5217 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
4520, 43fssresd 6586 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . . . . 15 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 22491 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4844, 45, 47syl2anc 587 . . . . . . . . . . . . 13 (𝜑X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4941, 48eqtr3id 2792 . . . . . . . . . . . 12 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝐿)
50 ptunhmeo.y . . . . . . . . . . . 12 𝑌 = 𝐿
5149, 50eqtr4di 2796 . . . . . . . . . . 11 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝑌)
5237, 51eqtrd 2777 . . . . . . . . . 10 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5352adantr 484 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5431, 53eleqtrrd 2841 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛))
5518adantr 484 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 8615 . . . . . . . 8 (((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) ∧ (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) ∧ 𝐴𝐶) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
5729, 54, 55, 56syl3anc 1373 . . . . . . 7 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
58 ixpfn 8584 . . . . . . 7 (((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
5957, 58syl 17 . . . . . 6 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
60 dffn5 6771 . . . . . 6 (((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶 ↔ ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6159, 60sylib 221 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6261mpteq2dva 5150 . . . 4 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
638, 62syl5eq 2790 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
64 ptunhmeo.j . . . 4 𝐽 = (∏t𝐹)
65 pttop 22479 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → (∏t‘(𝐹𝐴)) ∈ Top)
6619, 21, 65syl2anc 587 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐴)) ∈ Top)
6722, 66eqeltrid 2842 . . . . . 6 (𝜑𝐾 ∈ Top)
6826toptopon 21814 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑋))
6967, 68sylib 221 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑋))
70 pttop 22479 . . . . . . . 8 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → (∏t‘(𝐹𝐵)) ∈ Top)
7144, 45, 70syl2anc 587 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐵)) ∈ Top)
7246, 71eqeltrid 2842 . . . . . 6 (𝜑𝐿 ∈ Top)
7350toptopon 21814 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘𝑌))
7472, 73sylib 221 . . . . 5 (𝜑𝐿 ∈ (TopOn‘𝑌))
75 txtopon 22488 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7669, 74, 75syl2anc 587 . . . 4 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7717eleq2d 2823 . . . . . . 7 (𝜑 → (𝑘𝐶𝑘 ∈ (𝐴𝐵)))
7877biimpa 480 . . . . . 6 ((𝜑𝑘𝐶) → 𝑘 ∈ (𝐴𝐵))
79 elun 4063 . . . . . 6 (𝑘 ∈ (𝐴𝐵) ↔ (𝑘𝐴𝑘𝐵))
8078, 79sylib 221 . . . . 5 ((𝜑𝑘𝐶) → (𝑘𝐴𝑘𝐵))
81 ixpfn 8584 . . . . . . . . . . 11 ((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) → (1st𝑧) Fn 𝐴)
8229, 81syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8382adantlr 715 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8451adantr 484 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐵 (𝐹𝑛) = 𝑌)
8531, 84eleqtrrd 2841 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛))
86 ixpfn 8584 . . . . . . . . . . 11 ((2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛) → (2nd𝑧) Fn 𝐵)
8785, 86syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8887adantlr 715 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8933ad2antrr 726 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
90 simplr 769 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐴)
91 fvun1 6802 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐴)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9283, 88, 89, 90, 91syl112anc 1376 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9392mpteq2dva 5150 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)))
9476adantr 484 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
954mpompt 7324 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
9669adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐾 ∈ (TopOn‘𝑋))
9774adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐿 ∈ (TopOn‘𝑌))
9896, 97cnmpt1st 22565 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
9995, 98eqeltrid 2842 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
10019adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐴 ∈ V)
10121adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝐴):𝐴⟶Top)
102 simpr 488 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝑘𝐴)
10326, 22ptpjcn 22508 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top ∧ 𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
104100, 101, 102, 103syl3anc 1373 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
105 fvres 6736 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
106105adantl 485 . . . . . . . . . 10 ((𝜑𝑘𝐴) → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
107106oveq2d 7229 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝐾 Cn ((𝐹𝐴)‘𝑘)) = (𝐾 Cn (𝐹𝑘)))
108104, 107eleqtrd 2840 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn (𝐹𝑘)))
109 fveq1 6716 . . . . . . . 8 (𝑓 = (1st𝑧) → (𝑓𝑘) = ((1st𝑧)‘𝑘))
11094, 99, 96, 108, 109cnmpt11 22560 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11193, 110eqeltrd 2838 . . . . . 6 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11282adantlr 715 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
11387adantlr 715 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
11433ad2antrr 726 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
115 simplr 769 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐵)
116 fvun2 6803 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐵)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
117112, 113, 114, 115, 116syl112anc 1376 . . . . . . . 8 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
118117mpteq2dva 5150 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)))
11976adantr 484 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
1205mpompt 7324 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
12169adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐾 ∈ (TopOn‘𝑋))
12274adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐿 ∈ (TopOn‘𝑌))
123121, 122cnmpt2nd 22566 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
124120, 123eqeltrid 2842 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
12544adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐵 ∈ V)
12645adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝐵) → (𝐹𝐵):𝐵⟶Top)
127 simpr 488 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝑘𝐵)
12850, 46ptpjcn 22508 . . . . . . . . . 10 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top ∧ 𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
129125, 126, 127, 128syl3anc 1373 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
130 fvres 6736 . . . . . . . . . . 11 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
131130adantl 485 . . . . . . . . . 10 ((𝜑𝑘𝐵) → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
132131oveq2d 7229 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝐿 Cn ((𝐹𝐵)‘𝑘)) = (𝐿 Cn (𝐹𝑘)))
133129, 132eleqtrd 2840 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn (𝐹𝑘)))
134 fveq1 6716 . . . . . . . 8 (𝑓 = (2nd𝑧) → (𝑓𝑘) = ((2nd𝑧)‘𝑘))
135119, 124, 122, 133, 134cnmpt11 22560 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
136118, 135eqeltrd 2838 . . . . . 6 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
137111, 136jaodan 958 . . . . 5 ((𝜑 ∧ (𝑘𝐴𝑘𝐵)) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13880, 137syldan 594 . . . 4 ((𝜑𝑘𝐶) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13964, 76, 15, 20, 138ptcn 22524 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14063, 139eqeltrd 2838 . 2 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14126, 50, 64, 22, 46, 1, 15, 20, 17, 33ptuncnv 22704 . . 3 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
142 pttop 22479 . . . . . . 7 ((𝐶𝑉𝐹:𝐶⟶Top) → (∏t𝐹) ∈ Top)
14315, 20, 142syl2anc 587 . . . . . 6 (𝜑 → (∏t𝐹) ∈ Top)
14464, 143eqeltrid 2842 . . . . 5 (𝜑𝐽 ∈ Top)
145 eqid 2737 . . . . . 6 𝐽 = 𝐽
146145toptopon 21814 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
147144, 146sylib 221 . . . 4 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
148145, 64, 22ptrescn 22536 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐴𝐶) → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
14915, 20, 18, 148syl3anc 1373 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
150145, 64, 46ptrescn 22536 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐵𝐶) → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
15115, 20, 43, 150syl3anc 1373 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
152147, 149, 151cnmpt1t 22562 . . 3 (𝜑 → (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
153141, 152eqeltrd 2838 . 2 (𝜑𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
154 ishmeo 22656 . 2 (𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽) ↔ (𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽) ∧ 𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿))))
155140, 153, 154sylanbrc 586 1 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  Vcvv 3408  cdif 3863  cun 3864  cin 3865  wss 3866  c0 4237  cop 4547   cuni 4819  cmpt 5135   × cxp 5549  ccnv 5550  cres 5553   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  1st c1st 7759  2nd c2nd 7760  Xcixp 8578  tcpt 16943  Topctop 21790  TopOnctopon 21807   Cn ccn 22121   ×t ctx 22457  Homeochmeo 22650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-fin 8630  df-fi 9027  df-topgen 16948  df-pt 16949  df-top 21791  df-topon 21808  df-bases 21843  df-cn 22124  df-cnp 22125  df-tx 22459  df-hmeo 22652
This theorem is referenced by:  xpstopnlem1  22706  ptcmpfi  22710
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