Theorem xpstopnlem1 23774

Description: The function 𝐹 used in xpsval 17534 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 23556	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		eqid 2736	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
7		0ex 5242	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ V)
9	6, 8, 2	pt1hmeo 23771	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
10		hmeocn 23725	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
11		cntop2 23206	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12	9, 10, 11	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
13		toptopon2 22883	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
14	12, 13	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
15		eqid 2736	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨1o, 𝐾⟩}) = (∏t‘{⟨1o, 𝐾⟩})
16		1on 8417	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1o ∈ On)
18	15, 17, 3	pt1hmeo 23771	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})))
19		hmeocn 23725	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})) → (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1o, 𝐾⟩})))
20		cntop2 23206	. . . . . . . . . . . 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 22883	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨1o, 𝐾⟩}) ∈ Top ↔ (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1o, 𝐾⟩})))
23	21, 22	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1o, 𝐾⟩})))
24		txtopon 23556	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩}))))
26		opeq2 4817	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
27	26	sneqd 4579	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
28		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})
29		snex 5381	. . . . . . . . . . . . . . 15 ⊢ {⟨∅, 𝑥⟩} ∈ V
30	27, 28, 29	fvmpt 6947	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
31		opeq2 4817	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ⟨1o, 𝑧⟩ = ⟨1o, 𝑦⟩)
32	31	sneqd 4579	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → {⟨1o, 𝑧⟩} = {⟨1o, 𝑦⟩})
33		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩}) = (𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})
34		snex 5381	. . . . . . . . . . . . . . 15 ⊢ {⟨1o, 𝑦⟩} ∈ V
35	32, 33, 34	fvmpt 6947	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑌 → ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦) = {⟨1o, 𝑦⟩})
36		opeq12 4818	. . . . . . . . . . . . . 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 597	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
38	37	mpoeq3ia 7445	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
39		toponuni 22879	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
40	2, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
41		toponuni 22879	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
42	3, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
43		mpoeq12 7440	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
46		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
47		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝐾 = ∪ 𝐾
48	46, 47, 9, 18	txhmeo 23768	. . . . . . . . . . 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 2836	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
50		hmeocn 23725	. . . . . . . . . 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 23214	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
54		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
55	54	fmpo 8021	. . . . . . . 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 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
57	56	r19.21bi 3229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
58	57	r19.21bi 3229	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
59	58	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})))
60		eqidd 2737	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩))
61		vex 3433	. . . . . . . . 9 ⊢ 𝑥 ∈ V
62		vex 3433	. . . . . . . . 9 ⊢ 𝑦 ∈ V
63	61, 62	op1std 7952	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
64	61, 62	op2ndd 7953	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
65	63, 64	uneq12d 4109	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑥 ∪ 𝑦))
66	65	mpompt 7481	. . . . . 6 ⊢ (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦))
67	66	eqcomi 2745	. . . . 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 7952	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (1st ‘𝑧) = {⟨∅, 𝑥⟩})
70	29, 34	op2ndd 7953	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (2nd ‘𝑧) = {⟨1o, 𝑦⟩})
71	69, 70	uneq12d 4109	. . . . 5 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}))
72		df-pr 4570	. . . . 5 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
73	71, 72	eqtr4di 2789	. . . 4 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
74	59, 60, 68, 73	fmpoco 8045	. . 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 2790	. 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)))
76		eqid 2736	. . . . 5 ⊢ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}))
77		eqid 2736	. . . . 5 ⊢ ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))
78		eqid 2736	. . . . 5 ⊢ (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}) = (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})
79		eqid 2736	. . . . 5 ⊢ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}))
80		eqid 2736	. . . . 5 ⊢ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))
81		eqid 2736	. . . . 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 8418	. . . . . 6 ⊢ 2o ∈ On
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 2o ∈ On)
84		topontop 22878	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
85	2, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
86		topontop 22878	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
87	3, 86	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
88		xpscf 17529	. . . . . 6 ⊢ ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}:2o⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
89	85, 87, 88	sylanbrc 584	. . . . 5 ⊢ (𝜑 → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}:2o⟶Top)
90		df2o3 8413	. . . . . . 7 ⊢ 2o = {∅, 1o}
91		df-pr 4570	. . . . . . 7 ⊢ {∅, 1o} = ({∅} ∪ {1o})
92	90, 91	eqtri 2759	. . . . . 6 ⊢ 2o = ({∅} ∪ {1o})
93	92	a1i 11	. . . . 5 ⊢ (𝜑 → 2o = ({∅} ∪ {1o}))
94		1n0 8423	. . . . . . 7 ⊢ 1o ≠ ∅
95	94	necomi 2986	. . . . . 6 ⊢ ∅ ≠ 1o
96		disjsn2 4656	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
97	95, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ({∅} ∩ {1o}) = ∅)
98	76, 77, 78, 79, 80, 81, 83, 89, 93, 97	ptunhmeo 23773	. . . 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 17521	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o)
100	2, 3, 99	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o)
101	7	prid1 4706	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 1o}
102	101, 90	eleqtrri 2835	. . . . . . . . 9 ⊢ ∅ ∈ 2o
103		fnressn 7112	. . . . . . . . 9 ⊢ (({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o ∧ ∅ ∈ 2o) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩})
104	100, 102, 103	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩})
105		fvpr0o 17523	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅) = 𝐽)
106	2, 105	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅) = 𝐽)
107	106	opeq2d 4823	. . . . . . . . 9 ⊢ (𝜑 → ⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩ = ⟨∅, 𝐽⟩)
108	107	sneqd 4579	. . . . . . . 8 ⊢ (𝜑 → {⟨∅, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘∅)⟩} = {⟨∅, 𝐽⟩})
109	104, 108	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅}) = {⟨∅, 𝐽⟩})
110	109	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
111	110	unieqd 4863	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) = ∪ (∏t‘{⟨∅, 𝐽⟩}))
112		1oex 8415	. . . . . . . . . . 11 ⊢ 1o ∈ V
113	112	prid2 4707	. . . . . . . . . 10 ⊢ 1o ∈ {∅, 1o}
114	113, 90	eleqtrri 2835	. . . . . . . . 9 ⊢ 1o ∈ 2o
115		fnressn 7112	. . . . . . . . 9 ⊢ (({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} Fn 2o ∧ 1o ∈ 2o) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩})
116	100, 114, 115	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩})
117		fvpr1o 17524	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o) = 𝐾)
118	3, 117	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o) = 𝐾)
119	118	opeq2d 4823	. . . . . . . . 9 ⊢ (𝜑 → ⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩ = ⟨1o, 𝐾⟩)
120	119	sneqd 4579	. . . . . . . 8 ⊢ (𝜑 → {⟨1o, ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}‘1o)⟩} = {⟨1o, 𝐾⟩})
121	116, 120	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}) = {⟨1o, 𝐾⟩})
122	121	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = (∏t‘{⟨1o, 𝐾⟩}))
123	122	unieqd 4863	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o})) = ∪ (∏t‘{⟨1o, 𝐾⟩}))
124		eqidd 2737	. . . . 5 ⊢ (𝜑 → (𝑥 ∪ 𝑦) = (𝑥 ∪ 𝑦))
125	111, 123, 124	mpoeq123dv 7442	. . . 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 7385	. . . . 5 ⊢ (𝜑 → ((∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {∅})) ×t (∏t‘({⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩} ↾ {1o}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})))
127	126	oveq1d 7382	. . . 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 2850	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
129		hmeoco 23737	. . 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 585	. 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 2836	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429   ∪ cun 3887   ∩ cin 3888  ∅c0 4273  {csn 4567  {cpr 4569  ⟨cop 4573  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629   ↾ cres 5633   ∘ ccom 5635  Oncon0 6323   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  1_oc1o 8398  2_oc2o 8399  ∏_tcpt 17401  Topctop 22858  TopOnctopon 22875   Cn ccn 23189   ×_t ctx 23525  Homeochmeo 23718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-1o 8405  df-2o 8406  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-fin 8897  df-fi 9324  df-topgen 17406  df-pt 17407  df-top 22859  df-topon 22876  df-bases 22911  df-cn 23192  df-cnp 23193  df-tx 23527  df-hmeo 23720

This theorem is referenced by:  xpstopnlem2  23776