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Theorem pt1hmeo 21419
 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 5085 . . . . 5 ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2 pt1hmeo.a . . . . . . 7 (𝜑𝐴𝑉)
32adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑉)
4 sneq 4135 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
54xpeq1d 5062 . . . . . . . 8 (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6 opeq1 4340 . . . . . . . . 9 (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
76sneqd 4137 . . . . . . . 8 (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
85, 7eqeq12d 2625 . . . . . . 7 (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9 vex 3176 . . . . . . . 8 𝑘 ∈ V
10 vex 3176 . . . . . . . 8 𝑥 ∈ V
119, 10xpsn 6313 . . . . . . 7 ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
128, 11vtoclg 3239 . . . . . 6 (𝐴𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
133, 12syl 17 . . . . 5 ((𝜑𝑥𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
141, 13syl5eqr 2658 . . . 4 ((𝜑𝑥𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
1514mpteq2dva 4672 . . 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 4835 . . . . 5 {𝐴} ∈ V
1918a1i 11 . . . 4 (𝜑 → {𝐴} ∈ V)
20 topontop 20541 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2117, 20syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
222, 21fsnd 6091 . . . 4 (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
2317cnmptid 21274 . . . . . 6 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
2423adantr 480 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
25 elsni 4142 . . . . . . . 8 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2625fveq2d 6107 . . . . . . 7 (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27 fvsng 6352 . . . . . . . 8 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
282, 17, 27syl2anc 691 . . . . . . 7 (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
2926, 28sylan9eqr 2666 . . . . . 6 ((𝜑𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
3029oveq2d 6565 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
3124, 30eleqtrrd 2691 . . . 4 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
3216, 17, 19, 22, 31ptcn 21240 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
3315, 32eqeltrrd 2689 . 2 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
3514adantrr 749 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
3634, 35eqtr4d 2647 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37 simprl 790 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥𝑋)
3837adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥𝑋)
39 eqid 2610 . . . . . . . . . 10 (𝑘 ∈ {𝐴} ↦ 𝑥) = (𝑘 ∈ {𝐴} ↦ 𝑥)
4038, 39fmptd 6292 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
41 toponmax 20543 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4217, 41syl 17 . . . . . . . . . . 11 (𝜑𝑋𝐽)
4342adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋𝐽)
44 elmapg 7757 . . . . . . . . . 10 ((𝑋𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4543, 18, 44sylancl 693 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4640, 45mpbird 246 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}))
4736, 46eqeltrd 2688 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
4834fveq1d 6105 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
492adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴𝑉)
50 fvsng 6352 . . . . . . . . 9 ((𝐴𝑉𝑥𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5149, 37, 50syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5248, 51eqtr2d 2645 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦𝐴))
5347, 52jca 553 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴)))
54 simprr 792 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥 = (𝑦𝐴))
55 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
5642adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑋𝐽)
57 elmapg 7757 . . . . . . . . . . 11 ((𝑋𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5856, 18, 57sylancl 693 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5955, 58mpbid 221 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦:{𝐴}⟶𝑋)
60 snidg 4153 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
612, 60syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
6261adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴 ∈ {𝐴})
6359, 62ffvelrnd 6268 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦𝐴) ∈ 𝑋)
6454, 63eqeltrd 2688 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥𝑋)
652adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴𝑉)
66 fsn2g 6311 . . . . . . . . . . 11 (𝐴𝑉 → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6765, 66syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6859, 67mpbid 221 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩}))
6968simprd 478 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, (𝑦𝐴)⟩})
7054opeq2d 4347 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦𝐴)⟩)
7170sneqd 4137 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦𝐴)⟩})
7269, 71eqtr4d 2647 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
7364, 72jca 553 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}))
7453, 73impbida 873 . . . . 5 (𝜑 → ((𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))))
7574mptcnv 5453 . . . 4 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)))
76 xpsng 6312 . . . . . . . . . . 11 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
772, 17, 76syl2anc 691 . . . . . . . . . 10 (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
7877eqcomd 2616 . . . . . . . . 9 (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
7978fveq2d 6107 . . . . . . . 8 (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
8016, 79syl5eq 2656 . . . . . . 7 (𝜑𝐾 = (∏t‘({𝐴} × {𝐽})))
81 eqid 2610 . . . . . . . . 9 (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
8281pttoponconst 21210 . . . . . . . 8 (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8319, 17, 82syl2anc 691 . . . . . . 7 (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8480, 83eqeltrd 2688 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})))
85 toponuni 20542 . . . . . 6 (𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})) → (𝑋𝑚 {𝐴}) = 𝐾)
8684, 85syl 17 . . . . 5 (𝜑 → (𝑋𝑚 {𝐴}) = 𝐾)
8786mpteq1d 4666 . . . 4 (𝜑 → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)) = (𝑦 𝐾 ↦ (𝑦𝐴)))
8875, 87eqtrd 2644 . . 3 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 𝐾 ↦ (𝑦𝐴)))
89 eqid 2610 . . . . . 6 𝐾 = 𝐾
9089, 16ptpjcn 21224 . . . . 5 (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9118, 22, 61, 90mp3an2i 1421 . . . 4 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9228oveq2d 6565 . . . 4 (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
9391, 92eleqtrd 2690 . . 3 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn 𝐽))
9488, 93eqeltrd 2688 . 2 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
95 ishmeo 21372 . 2 ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
9633, 94, 95sylanbrc 695 1 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ∏tcpt 15922  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  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-hmeo 21368 This theorem is referenced by:  xpstopnlem1  21422  ptcmpfi  21426
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