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Theorem pt1hmeo 22406
Description: The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
pt1hmeo.j 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
pt1hmeo.a (𝜑𝐴𝑉)
pt1hmeo.r (𝜑𝐽 ∈ (TopOn‘𝑋))
Assertion
Ref Expression
pt1hmeo (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑋
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem pt1hmeo
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconstmpt 5607 . . . . 5 ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2 pt1hmeo.a . . . . . . 7 (𝜑𝐴𝑉)
32adantr 483 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑉)
4 sneq 4569 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
54xpeq1d 5577 . . . . . . . 8 (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6 opeq1 4795 . . . . . . . . 9 (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
76sneqd 4571 . . . . . . . 8 (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
85, 7eqeq12d 2835 . . . . . . 7 (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9 vex 3496 . . . . . . . 8 𝑘 ∈ V
10 vex 3496 . . . . . . . 8 𝑥 ∈ V
119, 10xpsn 6896 . . . . . . 7 ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
128, 11vtoclg 3566 . . . . . 6 (𝐴𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
133, 12syl 17 . . . . 5 ((𝜑𝑥𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
141, 13syl5eqr 2868 . . . 4 ((𝜑𝑥𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
1514mpteq2dva 5152 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) = (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}))
16 pt1hmeo.j . . . 4 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
17 pt1hmeo.r . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
18 snex 5322 . . . . 5 {𝐴} ∈ V
1918a1i 11 . . . 4 (𝜑 → {𝐴} ∈ V)
20 topontop 21513 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2117, 20syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
222, 21fsnd 6650 . . . 4 (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
2317cnmptid 22261 . . . . . 6 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
2423adantr 483 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
25 elsni 4576 . . . . . . . 8 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2625fveq2d 6667 . . . . . . 7 (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27 fvsng 6935 . . . . . . . 8 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
282, 17, 27syl2anc 586 . . . . . . 7 (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
2926, 28sylan9eqr 2876 . . . . . 6 ((𝜑𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
3029oveq2d 7164 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
3124, 30eleqtrrd 2914 . . . 4 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
3216, 17, 19, 22, 31ptcn 22227 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
3315, 32eqeltrrd 2912 . 2 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
3514adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
3634, 35eqtr4d 2857 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37 simprl 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥𝑋)
3837adantr 483 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥𝑋)
3938fmpttd 6872 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40 toponmax 21526 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4117, 40syl 17 . . . . . . . . . . 11 (𝜑𝑋𝐽)
4241adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋𝐽)
43 elmapg 8411 . . . . . . . . . 10 ((𝑋𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4442, 18, 43sylancl 588 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4539, 44mpbird 259 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}))
4636, 45eqeltrd 2911 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋m {𝐴}))
4734fveq1d 6665 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
482adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴𝑉)
49 fvsng 6935 . . . . . . . . 9 ((𝐴𝑉𝑥𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5048, 37, 49syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5147, 50eqtr2d 2855 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦𝐴))
5246, 51jca 514 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴)))
53 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥 = (𝑦𝐴))
54 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 ∈ (𝑋m {𝐴}))
5541adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑋𝐽)
56 elmapg 8411 . . . . . . . . . . 11 ((𝑋𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5755, 18, 56sylancl 588 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦 ∈ (𝑋m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5854, 57mpbid 234 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦:{𝐴}⟶𝑋)
59 snidg 4591 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
602, 59syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
6160adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴 ∈ {𝐴})
6258, 61ffvelrnd 6845 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦𝐴) ∈ 𝑋)
6353, 62eqeltrd 2911 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥𝑋)
642adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴𝑉)
65 fsn2g 6893 . . . . . . . . . . 11 (𝐴𝑉 → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6664, 65syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6758, 66mpbid 234 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩}))
6867simprd 498 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, (𝑦𝐴)⟩})
6953opeq2d 4802 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦𝐴)⟩)
7069sneqd 4571 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦𝐴)⟩})
7168, 70eqtr4d 2857 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
7263, 71jca 514 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}))
7352, 72impbida 799 . . . . 5 (𝜑 → ((𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))))
7473mptcnv 5991 . . . 4 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋m {𝐴}) ↦ (𝑦𝐴)))
75 xpsng 6894 . . . . . . . . . . 11 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
762, 17, 75syl2anc 586 . . . . . . . . . 10 (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
7776eqcomd 2825 . . . . . . . . 9 (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
7877fveq2d 6667 . . . . . . . 8 (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
7916, 78syl5eq 2866 . . . . . . 7 (𝜑𝐾 = (∏t‘({𝐴} × {𝐽})))
80 eqid 2819 . . . . . . . . 9 (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
8180pttoponconst 22197 . . . . . . . 8 (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋m {𝐴})))
8219, 17, 81syl2anc 586 . . . . . . 7 (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋m {𝐴})))
8379, 82eqeltrd 2911 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑋m {𝐴})))
84 toponuni 21514 . . . . . 6 (𝐾 ∈ (TopOn‘(𝑋m {𝐴})) → (𝑋m {𝐴}) = 𝐾)
8583, 84syl 17 . . . . 5 (𝜑 → (𝑋m {𝐴}) = 𝐾)
8685mpteq1d 5146 . . . 4 (𝜑 → (𝑦 ∈ (𝑋m {𝐴}) ↦ (𝑦𝐴)) = (𝑦 𝐾 ↦ (𝑦𝐴)))
8774, 86eqtrd 2854 . . 3 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 𝐾 ↦ (𝑦𝐴)))
88 eqid 2819 . . . . . 6 𝐾 = 𝐾
8988, 16ptpjcn 22211 . . . . 5 (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9018, 22, 60, 89mp3an2i 1459 . . . 4 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9128oveq2d 7164 . . . 4 (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
9290, 91eleqtrd 2913 . . 3 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn 𝐽))
9387, 92eqeltrd 2911 . 2 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94 ishmeo 22359 . 2 ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
9533, 93, 94sylanbrc 585 1 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  {csn 4559  cop 4565   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  wf 6344  cfv 6348  (class class class)co 7148  m cmap 8398  tcpt 16704  Topctop 21493  TopOnctopon 21510   Cn ccn 21824  Homeochmeo 22353
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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-fin 8505  df-fi 8867  df-topgen 16709  df-pt 16710  df-top 21494  df-topon 21511  df-bases 21546  df-cn 21827  df-cnp 21828  df-hmeo 22355
This theorem is referenced by:  xpstopnlem1  22409  ptcmpfi  22413
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