Theorem ptunhmeo 24035

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 (𝐴↑𝐵) · (𝐴↑𝐶) = 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 = (∏t‘𝐹)
ptunhmeo.k	⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))
ptunhmeo.l	⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))
ptunhmeo.g	⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
ptunhmeo.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
ptunhmeo.f	⊢ (𝜑 → 𝐹:𝐶⟶Top)
ptunhmeo.u	⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))
ptunhmeo.i	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
ptunhmeo	⊢ (𝜑 → 𝐺 ∈ ((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.

Step	Hyp	Ref	Expression
1		ptunhmeo.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
2		vex 3457	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	op1std 7999	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
5	2, 3	op2ndd 8000	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
6	4, 5	uneq12d 4119	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑥 ∪ 𝑦))
7	6	mpompt 7530	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
8	1, 7	eqtr4i 2788	. . . 4 ⊢ 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧)))
9		xp1st 8021	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
10	9	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st ‘𝑧) ∈ 𝑋)
11		ixpeq2 8921	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑛) = ∪ (𝐹‘𝑛) → X𝑛 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑛) = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
12		fvres 6901	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑛) = (𝐹‘𝑛))
13	12	unieqd 4883	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐴 → ∪ ((𝐹 ↾ 𝐴)‘𝑛) = ∪ (𝐹‘𝑛))
14	11, 13	mprg 3084	. . . . . . . . . . . 12 ⊢ X𝑛 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑛) = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
15		ptunhmeo.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
16		ssun1 4127	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
17		ptunhmeo.u	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))
18	16, 17	sseqtrrid 3977	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
19	15, 18	ssexd 5293	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ V)
20		ptunhmeo.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐶⟶Top)
21	20, 18	fssresd 6746	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶Top)
22		ptunhmeo.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))
23	22	ptuni 23821	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ V ∧ (𝐹 ↾ 𝐴):𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑛) = ∪ 𝐾)
24	19, 21, 23	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑛) = ∪ 𝐾)
25	14, 24	eqtr3id 2811	. . . . . . . . . . 11 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐾)
26		ptunhmeo.x	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐾
27	25, 26	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
28	27	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
29	10, 28	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st ‘𝑧) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
30		xp2nd 8022	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2nd ‘𝑧) ∈ 𝑌)
31	30	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) ∈ 𝑌)
32	17	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝐶)
33		ptunhmeo.i	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
34		uneqdifeq 4451	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))
35	18, 33, 34	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))
36	32, 35	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = 𝐵)
37	36	ixpeq1d 8919	. . . . . . . . . . 11 ⊢ (𝜑 → X𝑛 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑛) = X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛))
38		ixpeq2 8921	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑛) = ∪ (𝐹‘𝑛) → X𝑛 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑛) = X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛))
39		fvres 6901	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑛) = (𝐹‘𝑛))
40	39	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐵 → ∪ ((𝐹 ↾ 𝐵)‘𝑛) = ∪ (𝐹‘𝑛))
41	38, 40	mprg 3084	. . . . . . . . . . . . 13 ⊢ X𝑛 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑛) = X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛)
42		ssun2 4128	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
43	42, 17	sseqtrrid 3977	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
44	15, 43	ssexd 5293	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ V)
45	20, 43	fssresd 6746	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶Top)
46		ptunhmeo.l	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))
47	46	ptuni 23821	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → X𝑛 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑛) = ∪ 𝐿)
48	44, 45, 47	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑛 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑛) = ∪ 𝐿)
49	41, 48	eqtr3id 2811	. . . . . . . . . . . 12 ⊢ (𝜑 → X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛) = ∪ 𝐿)
50		ptunhmeo.y	. . . . . . . . . . . 12 ⊢ 𝑌 = ∪ 𝐿
51	49, 50	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝜑 → X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛) = 𝑌)
52	37, 51	eqtrd 2797	. . . . . . . . . 10 ⊢ (𝜑 → X𝑛 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑛) = 𝑌)
53	52	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑛) = 𝑌)
54	31, 53	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) ∈ X𝑛 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑛))
55	18	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝐴 ⊆ 𝐶)
56		undifixp 8944	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∧ (2nd ‘𝑧) ∈ X𝑛 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑛) ∧ 𝐴 ⊆ 𝐶) → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ X𝑛 ∈ 𝐶 ∪ (𝐹‘𝑛))
57	29, 54, 55, 56	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ X𝑛 ∈ 𝐶 ∪ (𝐹‘𝑛))
58		ixpfn 8913	. . . . . . 7 ⊢ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ X𝑛 ∈ 𝐶 ∪ (𝐹‘𝑛) → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) Fn 𝐶)
59	57, 58	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) Fn 𝐶)
60		dffn5 6940	. . . . . 6 ⊢ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) Fn 𝐶 ↔ ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑘 ∈ 𝐶 ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)))
61	59, 60	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑘 ∈ 𝐶 ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)))
62	61	mpteq2dva 5202	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘 ∈ 𝐶 ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘))))
63	8, 62	eqtrid 2809	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘 ∈ 𝐶 ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘))))
64		ptunhmeo.j	. . . 4 ⊢ 𝐽 = (∏t‘𝐹)
65		pttop 23809	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝐹 ↾ 𝐴):𝐴⟶Top) → (∏t‘(𝐹 ↾ 𝐴)) ∈ Top)
66	19, 21, 65	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (∏t‘(𝐹 ↾ 𝐴)) ∈ Top)
67	22, 66	eqeltrid 2866	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
68	26	toptopon 23143	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑋))
69	67, 68	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
70		pttop 23809	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
71	44, 45, 70	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (∏t‘(𝐹 ↾ 𝐵)) ∈ Top)
72	46, 71	eqeltrid 2866	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ Top)
73	50	toptopon 23143	. . . . . 6 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘𝑌))
74	72, 73	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))
75		txtopon 23818	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
76	69, 74, 75	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
77	17	eleq2d 2848	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))
78	77	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ (𝐴 ∪ 𝐵))
79		elun 4103	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
80	78, 79	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
81		ixpfn 8913	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → (1st ‘𝑧) Fn 𝐴)
82	29, 81	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st ‘𝑧) Fn 𝐴)
83	82	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st ‘𝑧) Fn 𝐴)
84	51	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛) = 𝑌)
85	31, 84	eleqtrrd 2865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) ∈ X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛))
86		ixpfn 8913	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) ∈ X𝑛 ∈ 𝐵 ∪ (𝐹‘𝑛) → (2nd ‘𝑧) Fn 𝐵)
87	85, 86	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) Fn 𝐵)
88	87	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) Fn 𝐵)
89	33	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴 ∩ 𝐵) = ∅)
90		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘 ∈ 𝐴)
91		fvun1 6973	. . . . . . . . 9 ⊢ (((1st ‘𝑧) Fn 𝐴 ∧ (2nd ‘𝑧) Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑘 ∈ 𝐴)) → (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘) = ((1st ‘𝑧)‘𝑘))
92	83, 88, 89, 90, 91	syl112anc 1401	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘) = ((1st ‘𝑧)‘𝑘))
93	92	mpteq2dva 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑧)‘𝑘)))
94	76	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
95	4	mpompt 7530	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
96	69	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ∈ (TopOn‘𝑋))
97	74	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐿 ∈ (TopOn‘𝑌))
98	96, 97	cnmpt1st 23895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
99	95, 98	eqeltrid 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐾))
100	19	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ V)
101	21	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹 ↾ 𝐴):𝐴⟶Top)
102		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
103	26, 22	ptpjcn 23838	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝐹 ↾ 𝐴):𝐴⟶Top ∧ 𝑘 ∈ 𝐴) → (𝑓 ∈ 𝑋 ↦ (𝑓‘𝑘)) ∈ (𝐾 Cn ((𝐹 ↾ 𝐴)‘𝑘)))
104	100, 101, 102, 103	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑓 ∈ 𝑋 ↦ (𝑓‘𝑘)) ∈ (𝐾 Cn ((𝐹 ↾ 𝐴)‘𝑘)))
105		fvres 6901	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑘) = (𝐹‘𝑘))
106	105	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑘) = (𝐹‘𝑘))
107	106	oveq2d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐾 Cn ((𝐹 ↾ 𝐴)‘𝑘)) = (𝐾 Cn (𝐹‘𝑘)))
108	104, 107	eleqtrd 2864	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑓 ∈ 𝑋 ↦ (𝑓‘𝑘)) ∈ (𝐾 Cn (𝐹‘𝑘)))
109		fveq1 6881	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑧) → (𝑓‘𝑘) = ((1st ‘𝑧)‘𝑘))
110	94, 99, 96, 108, 109	cnmpt11 23890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
111	93, 110	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
112	82	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (1st ‘𝑧) Fn 𝐴)
113	87	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑧) Fn 𝐵)
114	33	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (𝐴 ∩ 𝐵) = ∅)
115		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → 𝑘 ∈ 𝐵)
116		fvun2 6974	. . . . . . . . 9 ⊢ (((1st ‘𝑧) Fn 𝐴 ∧ (2nd ‘𝑧) Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑘 ∈ 𝐵)) → (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘) = ((2nd ‘𝑧)‘𝑘))
117	112, 113, 114, 115, 116	syl112anc 1401	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘) = ((2nd ‘𝑧)‘𝑘))
118	117	mpteq2dva 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ‘𝑧)‘𝑘)))
119	76	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
120	5	mpompt 7530	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
121	69	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐾 ∈ (TopOn‘𝑋))
122	74	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐿 ∈ (TopOn‘𝑌))
123	121, 122	cnmpt2nd 23896	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
124	120, 123	eqeltrid 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) ∈ ((𝐾 ×t 𝐿) Cn 𝐿))
125	44	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐵 ∈ V)
126	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝐹 ↾ 𝐵):𝐵⟶Top)
127		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
128	50, 46	ptpjcn 23838	. . . . . . . . . 10 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top ∧ 𝑘 ∈ 𝐵) → (𝑓 ∈ 𝑌 ↦ (𝑓‘𝑘)) ∈ (𝐿 Cn ((𝐹 ↾ 𝐵)‘𝑘)))
129	125, 126, 127, 128	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑓 ∈ 𝑌 ↦ (𝑓‘𝑘)) ∈ (𝐿 Cn ((𝐹 ↾ 𝐵)‘𝑘)))
130		fvres 6901	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
131	130	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
132	131	oveq2d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝐿 Cn ((𝐹 ↾ 𝐵)‘𝑘)) = (𝐿 Cn (𝐹‘𝑘)))
133	129, 132	eleqtrd 2864	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑓 ∈ 𝑌 ↦ (𝑓‘𝑘)) ∈ (𝐿 Cn (𝐹‘𝑘)))
134		fveq1 6881	. . . . . . . 8 ⊢ (𝑓 = (2nd ‘𝑧) → (𝑓‘𝑘) = ((2nd ‘𝑧)‘𝑘))
135	119, 124, 122, 133, 134	cnmpt11 23890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ‘𝑧)‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
136	118, 135	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
137	111, 136	jaodan 972	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
138	80, 137	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝑧 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘)) ∈ ((𝐾 ×t 𝐿) Cn (𝐹‘𝑘)))
139	64, 76, 15, 20, 138	ptcn 23854	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (𝑘 ∈ 𝐶 ↦ (((1st ‘𝑧) ∪ (2nd ‘𝑧))‘𝑘))) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
140	63, 139	eqeltrd 2862	. 2 ⊢ (𝜑 → 𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
141	26, 50, 64, 22, 46, 1, 15, 20, 17, 33	ptuncnv 24034	. . 3 ⊢ (𝜑 → ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩))
142		pttop 23809	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐹:𝐶⟶Top) → (∏t‘𝐹) ∈ Top)
143	15, 20, 142	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
144	64, 143	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
145		eqid 2762	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
146	145	toptopon 23143	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
147	144, 146	sylib 221	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
148	145, 64, 22	ptrescn 23866	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐹:𝐶⟶Top ∧ 𝐴 ⊆ 𝐶) → (𝑧 ∈ ∪ 𝐽 ↦ (𝑧 ↾ 𝐴)) ∈ (𝐽 Cn 𝐾))
149	15, 20, 18, 148	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐽 ↦ (𝑧 ↾ 𝐴)) ∈ (𝐽 Cn 𝐾))
150	145, 64, 46	ptrescn 23866	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐹:𝐶⟶Top ∧ 𝐵 ⊆ 𝐶) → (𝑧 ∈ ∪ 𝐽 ↦ (𝑧 ↾ 𝐵)) ∈ (𝐽 Cn 𝐿))
151	15, 20, 43, 150	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐽 ↦ (𝑧 ↾ 𝐵)) ∈ (𝐽 Cn 𝐿))
152	147, 149, 151	cnmpt1t 23892	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
153	141, 152	eqeltrd 2862	. 2 ⊢ (𝜑 → ◡𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
154		ishmeo 23986	. 2 ⊢ (𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽) ↔ (𝐺 ∈ ((𝐾 ×t 𝐿) Cn 𝐽) ∧ ◡𝐺 ∈ (𝐽 Cn (𝐾 ×t 𝐿))))
155	140, 153, 154	sylanbrc 595	1 ⊢ (𝜑 → 𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593  ∪ cuni 4870   ↦ cmpt 5190   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7987  2^nd c2nd 7988  Xcixp 8907  ∏_tcpt 17527  Topctop 23119  TopOnctopon 23136   Cn ccn 23450   ×_t ctx 23787  Homeochmeo 23980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-1o 8458  df-2o 8459  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-fin 8959  df-fi 9384  df-topgen 17532  df-pt 17533  df-top 23120  df-topon 23137  df-bases 23172  df-cn 23453  df-cnp 23454  df-tx 23789  df-hmeo 23982

This theorem is used by:  xpstopnlem1  24036  ptcmpfi  24040