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Theorem ptunhmeo 21521
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 3189 . . . . . . . 8 𝑥 ∈ V
3 vex 3189 . . . . . . . 8 𝑦 ∈ V
42, 3op1std 7123 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
52, 3op2ndd 7124 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
64, 5uneq12d 3746 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
76mpt2mpt 6705 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2646 . . . 4 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧)))
9 xp1st 7143 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
109adantl 482 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ 𝑋)
11 ixpeq2 7866 . . . . . . . . . . . . 13 (∀𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛))
12 fvres 6164 . . . . . . . . . . . . . 14 (𝑛𝐴 → ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1312unieqd 4412 . . . . . . . . . . . . 13 (𝑛𝐴 ((𝐹𝐴)‘𝑛) = (𝐹𝑛))
1411, 13mprg 2921 . . . . . . . . . . . 12 X𝑛𝐴 ((𝐹𝐴)‘𝑛) = X𝑛𝐴 (𝐹𝑛)
15 ptunhmeo.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑉)
16 ssun1 3754 . . . . . . . . . . . . . . 15 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . . . . 15 (𝜑𝐶 = (𝐴𝐵))
1816, 17syl5sseqr 3633 . . . . . . . . . . . . . 14 (𝜑𝐴𝐶)
1915, 18ssexd 4765 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6028 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . . . . 14 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 21307 . . . . . . . . . . . . 13 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2419, 21, 23syl2anc 692 . . . . . . . . . . . 12 (𝜑X𝑛𝐴 ((𝐹𝐴)‘𝑛) = 𝐾)
2514, 24syl5eqr 2669 . . . . . . . . . . 11 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝐾)
26 ptunhmeo.x . . . . . . . . . . 11 𝑋 = 𝐾
2725, 26syl6eqr 2673 . . . . . . . . . 10 (𝜑X𝑛𝐴 (𝐹𝑛) = 𝑋)
2827adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
2910, 28eleqtrrd 2701 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛))
30 xp2nd 7144 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
3130adantl 482 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ 𝑌)
3217eqcomd 2627 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . . . . 14 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4029 . . . . . . . . . . . . . 14 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 222 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 7864 . . . . . . . . . . 11 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = X𝑛𝐵 (𝐹𝑛))
38 ixpeq2 7866 . . . . . . . . . . . . . 14 (∀𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛))
39 fvres 6164 . . . . . . . . . . . . . . 15 (𝑛𝐵 → ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4039unieqd 4412 . . . . . . . . . . . . . 14 (𝑛𝐵 ((𝐹𝐵)‘𝑛) = (𝐹𝑛))
4138, 40mprg 2921 . . . . . . . . . . . . 13 X𝑛𝐵 ((𝐹𝐵)‘𝑛) = X𝑛𝐵 (𝐹𝑛)
42 ssun2 3755 . . . . . . . . . . . . . . . 16 𝐵 ⊆ (𝐴𝐵)
4342, 17syl5sseqr 3633 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
4415, 43ssexd 4765 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
4520, 43fssresd 6028 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . . . . 15 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 21307 . . . . . . . . . . . . . 14 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4844, 45, 47syl2anc 692 . . . . . . . . . . . . 13 (𝜑X𝑛𝐵 ((𝐹𝐵)‘𝑛) = 𝐿)
4941, 48syl5eqr 2669 . . . . . . . . . . . 12 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝐿)
50 ptunhmeo.y . . . . . . . . . . . 12 𝑌 = 𝐿
5149, 50syl6eqr 2673 . . . . . . . . . . 11 (𝜑X𝑛𝐵 (𝐹𝑛) = 𝑌)
5237, 51eqtrd 2655 . . . . . . . . . 10 (𝜑X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5352adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) = 𝑌)
5431, 53eleqtrrd 2701 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛))
5518adantr 481 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 7888 . . . . . . . 8 (((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) ∧ (2nd𝑧) ∈ X𝑛 ∈ (𝐶𝐴) (𝐹𝑛) ∧ 𝐴𝐶) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
5729, 54, 55, 56syl3anc 1323 . . . . . . 7 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛))
58 ixpfn 7858 . . . . . . 7 (((1st𝑧) ∪ (2nd𝑧)) ∈ X𝑛𝐶 (𝐹𝑛) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
5957, 58syl 17 . . . . . 6 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶)
60 dffn5 6198 . . . . . 6 (((1st𝑧) ∪ (2nd𝑧)) Fn 𝐶 ↔ ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6159, 60sylib 208 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → ((1st𝑧) ∪ (2nd𝑧)) = (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)))
6261mpteq2dva 4704 . . . 4 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
638, 62syl5eq 2667 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))))
64 ptunhmeo.j . . . 4 𝐽 = (∏t𝐹)
65 pttop 21295 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → (∏t‘(𝐹𝐴)) ∈ Top)
6619, 21, 65syl2anc 692 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐴)) ∈ Top)
6722, 66syl5eqel 2702 . . . . . 6 (𝜑𝐾 ∈ Top)
6826toptopon 20648 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑋))
6967, 68sylib 208 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑋))
70 pttop 21295 . . . . . . . 8 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → (∏t‘(𝐹𝐵)) ∈ Top)
7144, 45, 70syl2anc 692 . . . . . . 7 (𝜑 → (∏t‘(𝐹𝐵)) ∈ Top)
7246, 71syl5eqel 2702 . . . . . 6 (𝜑𝐿 ∈ Top)
7350toptopon 20648 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘𝑌))
7472, 73sylib 208 . . . . 5 (𝜑𝐿 ∈ (TopOn‘𝑌))
75 txtopon 21304 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7669, 74, 75syl2anc 692 . . . 4 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
7717eleq2d 2684 . . . . . . 7 (𝜑 → (𝑘𝐶𝑘 ∈ (𝐴𝐵)))
7877biimpa 501 . . . . . 6 ((𝜑𝑘𝐶) → 𝑘 ∈ (𝐴𝐵))
79 elun 3731 . . . . . 6 (𝑘 ∈ (𝐴𝐵) ↔ (𝑘𝐴𝑘𝐵))
8078, 79sylib 208 . . . . 5 ((𝜑𝑘𝐶) → (𝑘𝐴𝑘𝐵))
81 ixpfn 7858 . . . . . . . . . . 11 ((1st𝑧) ∈ X𝑛𝐴 (𝐹𝑛) → (1st𝑧) Fn 𝐴)
8229, 81syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8382adantlr 750 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
8451adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → X𝑛𝐵 (𝐹𝑛) = 𝑌)
8531, 84eleqtrrd 2701 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛))
86 ixpfn 7858 . . . . . . . . . . 11 ((2nd𝑧) ∈ X𝑛𝐵 (𝐹𝑛) → (2nd𝑧) Fn 𝐵)
8785, 86syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8887adantlr 750 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
8933ad2antrr 761 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
90 simplr 791 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐴)
91 fvun1 6226 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐴)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9283, 88, 89, 90, 91syl112anc 1327 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((1st𝑧)‘𝑘))
9392mpteq2dva 4704 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)))
9476adantr 481 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
954mpt2mpt 6705 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
9669adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐾 ∈ (TopOn‘𝑋))
9774adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐿 ∈ (TopOn‘𝑌))
9896, 97cnmpt1st 21381 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
9995, 98syl5eqel 2702 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
10019adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐴 ∈ V)
10121adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝐴):𝐴⟶Top)
102 simpr 477 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝑘𝐴)
10326, 22ptpjcn 21324 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top ∧ 𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
104100, 101, 102, 103syl3anc 1323 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn ((𝐹𝐴)‘𝑘)))
105 fvres 6164 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
106105adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝐴) → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
107106oveq2d 6620 . . . . . . . . 9 ((𝜑𝑘𝐴) → (𝐾 Cn ((𝐹𝐴)‘𝑘)) = (𝐾 Cn (𝐹𝑘)))
108104, 107eleqtrd 2700 . . . . . . . 8 ((𝜑𝑘𝐴) → (𝑓𝑋 ↦ (𝑓𝑘)) ∈ (𝐾 Cn (𝐹𝑘)))
109 fveq1 6147 . . . . . . . 8 (𝑓 = (1st𝑧) → (𝑓𝑘) = ((1st𝑧)‘𝑘))
11094, 99, 96, 108, 109cnmpt11 21376 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11193, 110eqeltrd 2698 . . . . . 6 ((𝜑𝑘𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
11282adantlr 750 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st𝑧) Fn 𝐴)
11387adantlr 750 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd𝑧) Fn 𝐵)
11433ad2antrr 761 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴𝐵) = ∅)
115 simplr 791 . . . . . . . . 9 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘𝐵)
116 fvun2 6227 . . . . . . . . 9 (((1st𝑧) Fn 𝐴 ∧ (2nd𝑧) Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑘𝐵)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
117112, 113, 114, 115, 116syl112anc 1327 . . . . . . . 8 (((𝜑𝑘𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st𝑧) ∪ (2nd𝑧))‘𝑘) = ((2nd𝑧)‘𝑘))
118117mpteq2dva 4704 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)))
11976adantr 481 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
1205mpt2mpt 6705 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
12169adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐾 ∈ (TopOn‘𝑋))
12274adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐿 ∈ (TopOn‘𝑌))
123121, 122cnmpt2nd 21382 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
124120, 123syl5eqel 2702 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
12544adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝐵 ∈ V)
12645adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐵) → (𝐹𝐵):𝐵⟶Top)
127 simpr 477 . . . . . . . . . 10 ((𝜑𝑘𝐵) → 𝑘𝐵)
12850, 46ptpjcn 21324 . . . . . . . . . 10 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top ∧ 𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
129125, 126, 127, 128syl3anc 1323 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn ((𝐹𝐵)‘𝑘)))
130 fvres 6164 . . . . . . . . . . 11 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
131130adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝐵) → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
132131oveq2d 6620 . . . . . . . . 9 ((𝜑𝑘𝐵) → (𝐿 Cn ((𝐹𝐵)‘𝑘)) = (𝐿 Cn (𝐹𝑘)))
133129, 132eleqtrd 2700 . . . . . . . 8 ((𝜑𝑘𝐵) → (𝑓𝑌 ↦ (𝑓𝑘)) ∈ (𝐿 Cn (𝐹𝑘)))
134 fveq1 6147 . . . . . . . 8 (𝑓 = (2nd𝑧) → (𝑓𝑘) = ((2nd𝑧)‘𝑘))
135119, 124, 122, 133, 134cnmpt11 21376 . . . . . . 7 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
136118, 135eqeltrd 2698 . . . . . 6 ((𝜑𝑘𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
137111, 136jaodan 825 . . . . 5 ((𝜑 ∧ (𝑘𝐴𝑘𝐵)) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13880, 137syldan 487 . . . 4 ((𝜑𝑘𝐶) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹𝑘)))
13964, 76, 15, 20, 138ptcn 21340 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘𝐶 ↦ (((1st𝑧) ∪ (2nd𝑧))‘𝑘))) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14063, 139eqeltrd 2698 . 2 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
14126, 50, 64, 22, 46, 1, 15, 20, 17, 33ptuncnv 21520 . . 3 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
142 pttop 21295 . . . . . . 7 ((𝐶𝑉𝐹:𝐶⟶Top) → (∏t𝐹) ∈ Top)
14315, 20, 142syl2anc 692 . . . . . 6 (𝜑 → (∏t𝐹) ∈ Top)
14464, 143syl5eqel 2702 . . . . 5 (𝜑𝐽 ∈ Top)
145 eqid 2621 . . . . . 6 𝐽 = 𝐽
146145toptopon 20648 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
147144, 146sylib 208 . . . 4 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
148145, 64, 22ptrescn 21352 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐴𝐶) → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
14915, 20, 18, 148syl3anc 1323 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐴)) ∈ (𝐽 Cn 𝐾))
150145, 64, 46ptrescn 21352 . . . . 5 ((𝐶𝑉𝐹:𝐶⟶Top ∧ 𝐵𝐶) → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
15115, 20, 43, 150syl3anc 1323 . . . 4 (𝜑 → (𝑧 𝐽 ↦ (𝑧𝐵)) ∈ (𝐽 Cn 𝐿))
152147, 149, 151cnmpt1t 21378 . . 3 (𝜑 → (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
153141, 152eqeltrd 2698 . 2 (𝜑𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
154 ishmeo 21472 . 2 (𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽) ↔ (𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽) ∧ 𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿))))
155140, 153, 154sylanbrc 697 1 (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  cop 4154   cuni 4402  cmpt 4673   × cxp 5072  ccnv 5073  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  Xcixp 7852  tcpt 16020  Topctop 20617  TopOnctopon 20618   Cn ccn 20938   ×t ctx 21273  Homeochmeo 21466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-topgen 16025  df-pt 16026  df-top 20621  df-bases 20622  df-topon 20623  df-cn 20941  df-cnp 20942  df-tx 21275  df-hmeo 21468
This theorem is referenced by:  xpstopnlem1  21522  ptcmpfi  21526
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