Theorem xpstopnlem1 23792

Description: The function 𝐹 used in xpsval 17525 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
xpstopnlem1.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpstopnlem1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
xpstopnlem1.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))

Assertion

Ref	Expression
xpstopnlem1	⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xpstopnlem1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpstopnlem1.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
2		xpstopnlem1.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		xpstopnlem1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 23574	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		eqid 2739	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
7		0ex 5229	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ V)
9	6, 8, 2	pt1hmeo 23789	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
10		hmeocn 23743	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
11		cntop2 23224	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12	9, 10, 11	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
13		toptopon2 22901	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
14	12, 13	sylib 219	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
15		eqid 2739	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨1o, 𝐾⟩}) = (∏t‘{⟨1o, 𝐾⟩})
16		1on 8407	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1o ∈ On)
18	15, 17, 3	pt1hmeo 23789	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})))
19		hmeocn 23743	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})) → (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1o, 𝐾⟩})))
20		cntop2 23224	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1o, 𝐾⟩})) → (∏t‘{⟨1o, 𝐾⟩}) ∈ Top)
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨1o, 𝐾⟩}) ∈ Top)
22		toptopon2 22901	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨1o, 𝐾⟩}) ∈ Top ↔ (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1o, 𝐾⟩})))
23	21, 22	sylib 219	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1o, 𝐾⟩})))
24		txtopon 23574	. . . . . . . . . 10 ⊢ (((∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})) ∧ (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1o, 𝐾⟩}))) → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩}))))
25	14, 23, 24	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩}))))
26		opeq2 4805	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
27	26	sneqd 4567	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
28		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})
29		snex 5368	. . . . . . . . . . . . . . 15 ⊢ {⟨∅, 𝑥⟩} ∈ V
30	27, 28, 29	fvmpt 6935	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
31		opeq2 4805	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ⟨1o, 𝑧⟩ = ⟨1o, 𝑦⟩)
32	31	sneqd 4567	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → {⟨1o, 𝑧⟩} = {⟨1o, 𝑦⟩})
33		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) = (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})
34		snex 5368	. . . . . . . . . . . . . . 15 ⊢ {⟨1o, 𝑦⟩} ∈ V
35	32, 33, 34	fvmpt 6935	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑌 → ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦) = {⟨1o, 𝑦⟩})
36		opeq12 4806	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩} ∧ ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦) = {⟨1o, 𝑦⟩}) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
37	30, 35, 36	syl2an 602	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
38	37	mpoeq3ia 7434	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
39		toponuni 22897	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
40	2, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
41		toponuni 22897	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
42	3, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
43		mpoeq12 7429	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ∪ 𝐽 ∧ 𝑌 = ∪ 𝐾) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
44	40, 42, 43	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
45	38, 44	eqtr3id 2788	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
46		eqid 2739	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
47		eqid 2739	. . . . . . . . . . . 12 ⊢ ∪ 𝐾 = ∪ 𝐾
48	46, 47, 9, 18	txhmeo 23786	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
49	45, 48	eqeltrd 2839	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
50		hmeocn 23743	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
51	49, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
52		cnf2 23232	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩}))) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
53	5, 25, 51, 52	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
54		eqid 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
55	54	fmpo 8010	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
56	53, 55	sylibr 235	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
57	56	r19.21bi 3231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
58	57	r19.21bi 3231	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
59	58	anasss 467	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
60		eqidd 2740	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩))
61		vex 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
62		vex 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
63	61, 62	op1std 7941	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
64	61, 62	op2ndd 7942	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
65	63, 64	uneq12d 4099	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑥 ∪ 𝑦))
66	65	mpompt 7470	. . . . . 6 ⊢ (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦))
67	66	eqcomi 2748	. . . . 5 ⊢ (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) = (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧)))
68	67	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) = (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))))
69	29, 34	op1std 7941	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (1st ‘𝑧) = {⟨∅, 𝑥⟩})
70	29, 34	op2ndd 7942	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (2nd ‘𝑧) = {⟨1o, 𝑦⟩})
71	69, 70	uneq12d 4099	. . . . 5 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}))
72		df-pr 4558	. . . . 5 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
73	71, 72	eqtr4di 2792	. . . 4 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
74	59, 60, 68, 73	fmpoco 8034	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
75	1, 74	eqtr4id 2793	. 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)))
76		eqid 2739	. . . . 5 ⊢ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}))
77		eqid 2739	. . . . 5 ⊢ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))
78		eqid 2739	. . . . 5 ⊢ (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}) = (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})
79		eqid 2739	. . . . 5 ⊢ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}))
80		eqid 2739	. . . . 5 ⊢ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))
81		eqid 2739	. . . . 5 ⊢ (𝑥 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})), 𝑦 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) ↦ (𝑥 ∪ 𝑦)) = (𝑥 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})), 𝑦 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) ↦ (𝑥 ∪ 𝑦))
82		2on 8408	. . . . . 6 ⊢ 2o ∈ On
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 2o ∈ On)
84		topontop 22896	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
85	2, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
86		topontop 22896	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
87	3, 86	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
88		xpscf 17520	. . . . . 6 ⊢ ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}:2o⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
89	85, 87, 88	sylanbrc 589	. . . . 5 ⊢ (𝜑 → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}:2o⟶Top)
90		df2o3 8403	. . . . . . 7 ⊢ 2o = {∅, 1o}
91		df-pr 4558	. . . . . . 7 ⊢ {∅, 1o} = ({∅} ∪ {1o})
92	90, 91	eqtri 2762	. . . . . 6 ⊢ 2o = ({∅} ∪ {1o})
93	92	a1i 11	. . . . 5 ⊢ (𝜑 → 2o = ({∅} ∪ {1o}))
94		1n0 8413	. . . . . . 7 ⊢ 1o ≠ ∅
95	94	necomi 2988	. . . . . 6 ⊢ ∅ ≠ 1o
96		disjsn2 4644	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
97	95, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ({∅} ∩ {1o}) = ∅)
98	76, 77, 78, 79, 80, 81, 83, 89, 93, 97	ptunhmeo 23791	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})), 𝑦 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) ×t (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
99		fnpr2o 17512	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o)
100	2, 3, 99	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o)
101	7	prid1 4694	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 1o}
102	101, 90	eleqtrri 2838	. . . . . . . . 9 ⊢ ∅ ∈ 2o
103		fnressn 7101	. . . . . . . . 9 ⊢ (({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o ∧ ∅ ∈ 2o) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩})
104	100, 102, 103	sylancl 592	. . . . . . . 8 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩})
105		fvpr0o 17514	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅) = 𝐽)
106	2, 105	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅) = 𝐽)
107	106	opeq2d 4811	. . . . . . . . 9 ⊢ (𝜑 → ⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩ = ⟨∅, 𝐽⟩)
108	107	sneqd 4567	. . . . . . . 8 ⊢ (𝜑 → {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩} = {⟨∅, 𝐽⟩})
109	104, 108	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, 𝐽⟩})
110	109	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
111	110	unieqd 4851	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = ∪ (∏t‘{⟨∅, 𝐽⟩}))
112		1oex 8405	. . . . . . . . . . 11 ⊢ 1o ∈ V
113	112	prid2 4695	. . . . . . . . . 10 ⊢ 1o ∈ {∅, 1o}
114	113, 90	eleqtrri 2838	. . . . . . . . 9 ⊢ 1o ∈ 2o
115		fnressn 7101	. . . . . . . . 9 ⊢ (({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o ∧ 1o ∈ 2o) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩})
116	100, 114, 115	sylancl 592	. . . . . . . 8 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩})
117		fvpr1o 17515	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o) = 𝐾)
118	3, 117	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o) = 𝐾)
119	118	opeq2d 4811	. . . . . . . . 9 ⊢ (𝜑 → ⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩ = ⟨1o, 𝐾⟩)
120	119	sneqd 4567	. . . . . . . 8 ⊢ (𝜑 → {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩} = {⟨1o, 𝐾⟩})
121	116, 120	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, 𝐾⟩})
122	121	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = (∏t‘{⟨1o, 𝐾⟩}))
123	122	unieqd 4851	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = ∪ (∏t‘{⟨1o, 𝐾⟩}))
124		eqidd 2740	. . . . 5 ⊢ (𝜑 → (𝑥 ∪ 𝑦) = (𝑥 ∪ 𝑦))
125	111, 123, 124	mpoeq123dv 7431	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})), 𝑦 ∈ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) ↦ (𝑥 ∪ 𝑦)) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)))
126	110, 122	oveq12d 7374	. . . . 5 ⊢ (𝜑 → ((∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) ×t (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})))
127	126	oveq1d 7371	. . . 4 ⊢ (𝜑 → (((∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) ×t (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})) = (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
128	98, 125, 127	3eltr3d 2853	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
129		hmeoco 23755	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))) ∧ (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}))) → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
130	49, 128, 129	syl2anc 590	. 2 ⊢ (𝜑 → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
131	75, 130	eqeltrd 2839	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431   ∪ cun 3881   ∩ cin 3882  ∅c0 4261  {csn 4555  {cpr 4557  ⟨cop 4561  ∪ cuni 4838   ↦ cmpt 5153   × cxp 5616   ↾ cres 5620   ∘ ccom 5622  Oncon0 6310   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  1_oc1o 8388  2_oc2o 8389  ∏_tcpt 17392  Topctop 22876  TopOnctopon 22893   Cn ccn 23207   ×_t ctx 23543  Homeochmeo 23736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-1o 8395  df-2o 8396  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9314  df-topgen 17397  df-pt 17398  df-top 22877  df-topon 22894  df-bases 22929  df-cn 23210  df-cnp 23211  df-tx 23545  df-hmeo 23738

This theorem is referenced by:  xpstopnlem2  23794