cvmlift2lem9 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem9 32671

Description: Lemma for cvmlift2 32676. (Contributed by Mario Carneiro, 1-Jun-2015.)

**Hypotheses**
Ref	Expression
cvmlift2.b	⊢ 𝐵 = ∪ 𝐶
cvmlift2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2.g	⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
cvmlift2.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift2.i	⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
cvmlift2.h	⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
cvmlift2.k	⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))
cvmlift2lem10.s	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾_t 𝑐)Homeo(𝐽 ↾_t 𝑘))))})
cvmlift2lem9.1	⊢ (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)
cvmlift2lem9.2	⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝑀))
cvmlift2lem9.3	⊢ (𝜑 → 𝑈 ∈ II)
cvmlift2lem9.4	⊢ (𝜑 → 𝑉 ∈ II)
cvmlift2lem9.5	⊢ (𝜑 → (II ↾_t 𝑈) ∈ Conn)
cvmlift2lem9.6	⊢ (𝜑 → (II ↾_t 𝑉) ∈ Conn)
cvmlift2lem9.7	⊢ (𝜑 → 𝑋 ∈ 𝑈)
cvmlift2lem9.8	⊢ (𝜑 → 𝑌 ∈ 𝑉)
cvmlift2lem9.9	⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (^◡𝐺 “ 𝑀))
cvmlift2lem9.10	⊢ (𝜑 → 𝑍 ∈ 𝑉)
cvmlift2lem9.11	⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn 𝐶))
cvmlift2lem9.w	⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝑋𝐾𝑌) ∈ 𝑏)

**Assertion**
Ref	Expression
cvmlift2lem9	⊢ (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))

Distinct variable groups: 𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑥,𝑦,𝑧,𝐹 𝜑,𝑏,𝑓,𝑥,𝑦,𝑧 𝑀,𝑏,𝑐,𝑑,𝑘,𝑠,𝑥,𝑦,𝑧 𝑆,𝑏,𝑓,𝑥,𝑦,𝑧 𝐽,𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑥,𝑦,𝑧 𝑇,𝑏,𝑐,𝑑,𝑠 𝑧,𝑈 𝐺,𝑏,𝑐,𝑓,𝑘,𝑥,𝑦,𝑧 𝑊,𝑐,𝑑 𝐻,𝑏,𝑐,𝑓,𝑥,𝑦,𝑧 𝑋,𝑏,𝑐,𝑑,𝑓,𝑘,𝑥,𝑦,𝑧 𝑧,𝑍 𝐶,𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑥,𝑦,𝑧 𝑃,𝑓,𝑘,𝑥,𝑦,𝑧 𝐵,𝑏,𝑐,𝑑,𝑥,𝑦,𝑧 𝑌,𝑏,𝑐,𝑑,𝑓,𝑘,𝑥,𝑦,𝑧 𝐾,𝑏,𝑐,𝑑,𝑓,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑘,𝑠,𝑐,𝑑) 𝐵(𝑓,𝑘,𝑠) 𝑃(𝑠,𝑏,𝑐,𝑑) 𝑆(𝑘,𝑠,𝑐,𝑑) 𝑇(𝑥,𝑦,𝑧,𝑓,𝑘) 𝑈(𝑥,𝑦,𝑓,𝑘,𝑠,𝑏,𝑐,𝑑) 𝐺(𝑠,𝑑) 𝐻(𝑘,𝑠,𝑑) 𝐾(𝑘,𝑠) 𝑀(𝑓) 𝑉(𝑥,𝑦,𝑧,𝑓,𝑘,𝑠,𝑏,𝑐,𝑑) 𝑊(𝑥,𝑦,𝑧,𝑓,𝑘,𝑠,𝑏) 𝑋(𝑠) 𝑌(𝑠) 𝑍(𝑥,𝑦,𝑓,𝑘,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift2lem9 Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2.b	. 2 ⊢ 𝐵 = ∪ 𝐶
2		iitop 23485	. . 3 ⊢ II ∈ Top
3		iiuni 23486	. . 3 ⊢ (0[,]1) = ∪ II
4	2, 2, 3, 3	txunii 22198	. 2 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
5		cvmlift2lem10.s	. 2 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾_t 𝑐)Homeo(𝐽 ↾_t 𝑘))))})
6		cvmlift2.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
7		cvmlift2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
8		cvmlift2.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9		cvmlift2.i	. . 3 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
10		cvmlift2.h	. . 3 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
11		cvmlift2.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))
12	1, 6, 7, 8, 9, 10, 11	cvmlift2lem5 32667	. 2 ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
13	1, 6, 7, 8, 9, 10, 11	cvmlift2lem7 32669	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺)
14	13, 7	eqeltrd 2890	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐾) ∈ ((II ×_t II) Cn 𝐽))
15	2, 2	txtopi 22195	. . 3 ⊢ (II ×_t II) ∈ Top
16	15	a1i 11	. 2 ⊢ (𝜑 → (II ×_t II) ∈ Top)
17		cvmlift2lem9.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ II)
18		elssuni 4830	. . . . . 6 ⊢ (𝑈 ∈ II → 𝑈 ⊆ ∪ II)
19	18, 3	sseqtrrdi 3966	. . . . 5 ⊢ (𝑈 ∈ II → 𝑈 ⊆ (0[,]1))
20	17, 19	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ (0[,]1))
21		cvmlift2lem9.7	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
22	20, 21	sseldd 3916	. . 3 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
23		cvmlift2lem9.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ II)
24		elssuni 4830	. . . . . 6 ⊢ (𝑉 ∈ II → 𝑉 ⊆ ∪ II)
25	24, 3	sseqtrrdi 3966	. . . . 5 ⊢ (𝑉 ∈ II → 𝑉 ⊆ (0[,]1))
26	23, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝑉 ⊆ (0[,]1))
27		cvmlift2lem9.8	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
28	26, 27	sseldd 3916	. . 3 ⊢ (𝜑 → 𝑌 ∈ (0[,]1))
29		opelxpi 5556	. . 3 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
30	22, 28, 29	syl2anc 587	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
31		cvmlift2lem9.2	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝑀))
32	12, 22, 28	fovrnd 7300	. . . 4 ⊢ (𝜑 → (𝑋𝐾𝑌) ∈ 𝐵)
33		fvco3 6737	. . . . . . . 8 ⊢ ((𝐾:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐾)‘⟨𝑋, 𝑌⟩) = (𝐹‘(𝐾‘⟨𝑋, 𝑌⟩)))
34	12, 30, 33	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐾)‘⟨𝑋, 𝑌⟩) = (𝐹‘(𝐾‘⟨𝑋, 𝑌⟩)))
35	13	fveq1d 6647	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐾)‘⟨𝑋, 𝑌⟩) = (𝐺‘⟨𝑋, 𝑌⟩))
36	34, 35	eqtr3d 2835	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐾‘⟨𝑋, 𝑌⟩)) = (𝐺‘⟨𝑋, 𝑌⟩))
37		df-ov 7138	. . . . . . 7 ⊢ (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
38	37	fveq2i 6648	. . . . . 6 ⊢ (𝐹‘(𝑋𝐾𝑌)) = (𝐹‘(𝐾‘⟨𝑋, 𝑌⟩))
39		df-ov 7138	. . . . . 6 ⊢ (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
40	36, 38, 39	3eqtr4g 2858	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝑋𝐾𝑌)) = (𝑋𝐺𝑌))
41		cvmlift2lem9.1	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)
42	40, 41	eqeltrd 2890	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑋𝐾𝑌)) ∈ 𝑀)
43		cvmlift2lem9.w	. . . . 5 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝑋𝐾𝑌) ∈ 𝑏)
44	5, 1, 43	cvmsiota 32637	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑀) ∧ (𝑋𝐾𝑌) ∈ 𝐵 ∧ (𝐹‘(𝑋𝐾𝑌)) ∈ 𝑀)) → (𝑊 ∈ 𝑇 ∧ (𝑋𝐾𝑌) ∈ 𝑊))
45	6, 31, 32, 42, 44	syl13anc 1369	. . 3 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝑋𝐾𝑌) ∈ 𝑊))
46	37	eleq1i 2880	. . . 4 ⊢ ((𝑋𝐾𝑌) ∈ 𝑊 ↔ (𝐾‘⟨𝑋, 𝑌⟩) ∈ 𝑊)
47	46	anbi2i 625	. . 3 ⊢ ((𝑊 ∈ 𝑇 ∧ (𝑋𝐾𝑌) ∈ 𝑊) ↔ (𝑊 ∈ 𝑇 ∧ (𝐾‘⟨𝑋, 𝑌⟩) ∈ 𝑊))
48	45, 47	sylib 221	. 2 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐾‘⟨𝑋, 𝑌⟩) ∈ 𝑊))
49		xpss12 5534	. . 3 ⊢ ((𝑈 ⊆ (0[,]1) ∧ 𝑉 ⊆ (0[,]1)) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
50	20, 26, 49	syl2anc 587	. 2 ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
51		snidg 4559	. . . . . . 7 ⊢ (𝑚 ∈ 𝑈 → 𝑚 ∈ {𝑚})
52	51	ad2antrl 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑚 ∈ {𝑚})
53		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑛 ∈ 𝑉)
54		ovres 7294	. . . . . 6 ⊢ ((𝑚 ∈ {𝑚} ∧ 𝑛 ∈ 𝑉) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑛) = (𝑚𝐾𝑛))
55	52, 53, 54	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑛) = (𝑚𝐾𝑛))
56		eqid 2798	. . . . . . . 8 ⊢ ∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉)) = ∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉))
57	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → II ∈ Top)
58		snex 5297	. . . . . . . . . . 11 ⊢ {𝑚} ∈ V
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → {𝑚} ∈ V)
60	23	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑉 ∈ II)
61		txrest 22236	. . . . . . . . . 10 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ ({𝑚} ∈ V ∧ 𝑉 ∈ II)) → ((II ×_t II) ↾_t ({𝑚} × 𝑉)) = ((II ↾_t {𝑚}) ×_t (II ↾_t 𝑉)))
62	57, 57, 59, 60, 61	syl22anc 837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((II ×_t II) ↾_t ({𝑚} × 𝑉)) = ((II ↾_t {𝑚}) ×_t (II ↾_t 𝑉)))
63		iitopon 23484	. . . . . . . . . . . 12 ⊢ II ∈ (TopOn‘(0[,]1))
64	20	sselda 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑈) → 𝑚 ∈ (0[,]1))
65	64	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑚 ∈ (0[,]1))
66		restsn2 21776	. . . . . . . . . . . 12 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝑚 ∈ (0[,]1)) → (II ↾_t {𝑚}) = 𝒫 {𝑚})
67	63, 65, 66	sylancr 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (II ↾_t {𝑚}) = 𝒫 {𝑚})
68		pwsn 4792	. . . . . . . . . . . 12 ⊢ 𝒫 {𝑚} = {∅, {𝑚}}
69		indisconn 22023	. . . . . . . . . . . 12 ⊢ {∅, {𝑚}} ∈ Conn
70	68, 69	eqeltri 2886	. . . . . . . . . . 11 ⊢ 𝒫 {𝑚} ∈ Conn
71	67, 70	eqeltrdi 2898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (II ↾_t {𝑚}) ∈ Conn)
72		cvmlift2lem9.6	. . . . . . . . . . 11 ⊢ (𝜑 → (II ↾_t 𝑉) ∈ Conn)
73	72	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (II ↾_t 𝑉) ∈ Conn)
74		txconn 22294	. . . . . . . . . 10 ⊢ (((II ↾_t {𝑚}) ∈ Conn ∧ (II ↾_t 𝑉) ∈ Conn) → ((II ↾_t {𝑚}) ×_t (II ↾_t 𝑉)) ∈ Conn)
75	71, 73, 74	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((II ↾_t {𝑚}) ×_t (II ↾_t 𝑉)) ∈ Conn)
76	62, 75	eqeltrd 2890	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((II ×_t II) ↾_t ({𝑚} × 𝑉)) ∈ Conn)
77	1, 6, 7, 8, 9, 10, 11	cvmlift2lem6 32668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0[,]1)) → (𝐾 ↾ ({𝑚} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑚} × (0[,]1))) Cn 𝐶))
78	65, 77	syldan 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ ({𝑚} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑚} × (0[,]1))) Cn 𝐶))
79	26	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑉 ⊆ (0[,]1))
80		xpss2 5539	. . . . . . . . . . . . 13 ⊢ (𝑉 ⊆ (0[,]1) → ({𝑚} × 𝑉) ⊆ ({𝑚} × (0[,]1)))
81	79, 80	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × 𝑉) ⊆ ({𝑚} × (0[,]1)))
82	65	snssd 4702	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → {𝑚} ⊆ (0[,]1))
83		xpss1 5538	. . . . . . . . . . . . . 14 ⊢ ({𝑚} ⊆ (0[,]1) → ({𝑚} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1)))
84	82, 83	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1)))
85	4	restuni 21767	. . . . . . . . . . . . 13 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑚} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1))) → ({𝑚} × (0[,]1)) = ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1))))
86	15, 84, 85	sylancr 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × (0[,]1)) = ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1))))
87	81, 86	sseqtrd 3955	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × 𝑉) ⊆ ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1))))
88		eqid 2798	. . . . . . . . . . . 12 ⊢ ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1))) = ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1)))
89	88	cnrest 21890	. . . . . . . . . . 11 ⊢ (((𝐾 ↾ ({𝑚} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑚} × (0[,]1))) Cn 𝐶) ∧ ({𝑚} × 𝑉) ⊆ ∪ ((II ×_t II) ↾_t ({𝑚} × (0[,]1)))) → ((𝐾 ↾ ({𝑚} × (0[,]1))) ↾ ({𝑚} × 𝑉)) ∈ ((((II ×_t II) ↾_t ({𝑚} × (0[,]1))) ↾_t ({𝑚} × 𝑉)) Cn 𝐶))
90	78, 87, 89	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × (0[,]1))) ↾ ({𝑚} × 𝑉)) ∈ ((((II ×_t II) ↾_t ({𝑚} × (0[,]1))) ↾_t ({𝑚} × 𝑉)) Cn 𝐶))
91	81	resabs1d 5849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × (0[,]1))) ↾ ({𝑚} × 𝑉)) = (𝐾 ↾ ({𝑚} × 𝑉)))
92	15	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (II ×_t II) ∈ Top)
93		ovex 7168	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ∈ V
94	58, 93	xpex 7456	. . . . . . . . . . . . 13 ⊢ ({𝑚} × (0[,]1)) ∈ V
95	94	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × (0[,]1)) ∈ V)
96		restabs 21770	. . . . . . . . . . . 12 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑚} × 𝑉) ⊆ ({𝑚} × (0[,]1)) ∧ ({𝑚} × (0[,]1)) ∈ V) → (((II ×_t II) ↾_t ({𝑚} × (0[,]1))) ↾_t ({𝑚} × 𝑉)) = ((II ×_t II) ↾_t ({𝑚} × 𝑉)))
97	92, 81, 95, 96	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (((II ×_t II) ↾_t ({𝑚} × (0[,]1))) ↾_t ({𝑚} × 𝑉)) = ((II ×_t II) ↾_t ({𝑚} × 𝑉)))
98	97	oveq1d 7150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((((II ×_t II) ↾_t ({𝑚} × (0[,]1))) ↾_t ({𝑚} × 𝑉)) Cn 𝐶) = (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn 𝐶))
99	90, 91, 98	3eltr3d 2904	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn 𝐶))
100		cvmtop1 32620	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
101	6, 100	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ Top)
102	101	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝐶 ∈ Top)
103	1	toptopon 21522	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
104	102, 103	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝐶 ∈ (TopOn‘𝐵))
105		df-ima 5532	. . . . . . . . . . 11 ⊢ (𝐾 “ ({𝑚} × 𝑉)) = ran (𝐾 ↾ ({𝑚} × 𝑉))
106		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑚 ∈ 𝑈)
107	106	snssd 4702	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → {𝑚} ⊆ 𝑈)
108		xpss1 5538	. . . . . . . . . . . . 13 ⊢ ({𝑚} ⊆ 𝑈 → ({𝑚} × 𝑉) ⊆ (𝑈 × 𝑉))
109		imass2 5932	. . . . . . . . . . . . 13 ⊢ (({𝑚} × 𝑉) ⊆ (𝑈 × 𝑉) → (𝐾 “ ({𝑚} × 𝑉)) ⊆ (𝐾 “ (𝑈 × 𝑉)))
110	107, 108, 109	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 “ ({𝑚} × 𝑉)) ⊆ (𝐾 “ (𝑈 × 𝑉)))
111		cvmlift2lem9.9	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (^◡𝐺 “ 𝑀))
112		imaco 6071	. . . . . . . . . . . . . . . 16 ⊢ ((^◡𝐾 ∘ ^◡𝐹) “ 𝑀) = (^◡𝐾 “ (^◡𝐹 “ 𝑀))
113		cnvco 5720	. . . . . . . . . . . . . . . . . 18 ⊢ ^◡(𝐹 ∘ 𝐾) = (^◡𝐾 ∘ ^◡𝐹)
114	13	cnveqd 5710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡(𝐹 ∘ 𝐾) = ^◡𝐺)
115	113, 114	syl5eqr 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (^◡𝐾 ∘ ^◡𝐹) = ^◡𝐺)
116	115	imaeq1d 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((^◡𝐾 ∘ ^◡𝐹) “ 𝑀) = (^◡𝐺 “ 𝑀))
117	112, 116	syl5eqr 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐾 “ (^◡𝐹 “ 𝑀)) = (^◡𝐺 “ 𝑀))
118	111, 117	sseqtrrd 3956	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (^◡𝐾 “ (^◡𝐹 “ 𝑀)))
119	12	ffund 6491	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐾)
120	12	fdmd 6497	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ((0[,]1) × (0[,]1)))
121	50, 120	sseqtrrd 3956	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ dom 𝐾)
122		funimass3 6801	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐾 ∧ (𝑈 × 𝑉) ⊆ dom 𝐾) → ((𝐾 “ (𝑈 × 𝑉)) ⊆ (^◡𝐹 “ 𝑀) ↔ (𝑈 × 𝑉) ⊆ (^◡𝐾 “ (^◡𝐹 “ 𝑀))))
123	119, 121, 122	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 “ (𝑈 × 𝑉)) ⊆ (^◡𝐹 “ 𝑀) ↔ (𝑈 × 𝑉) ⊆ (^◡𝐾 “ (^◡𝐹 “ 𝑀))))
124	118, 123	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 “ (𝑈 × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
125	124	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 “ (𝑈 × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
126	110, 125	sstrd 3925	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 “ ({𝑚} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
127	105, 126	eqsstrrid 3964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ran (𝐾 ↾ ({𝑚} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
128		cnvimass 5916	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ 𝑀) ⊆ dom 𝐹
129		cvmcn 32622	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
130	6, 129	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
131		eqid 2798	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
132	1, 131	cnf 21851	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
133		fdm 6495	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐵⟶∪ 𝐽 → dom 𝐹 = 𝐵)
134	130, 132, 133	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐵)
135	128, 134	sseqtrid 3967	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ 𝑀) ⊆ 𝐵)
136	135	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (^◡𝐹 “ 𝑀) ⊆ 𝐵)
137		cnrest2 21891	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran (𝐾 ↾ ({𝑚} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀) ∧ (^◡𝐹 “ 𝑀) ⊆ 𝐵) → ((𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn 𝐶) ↔ (𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
138	104, 127, 136, 137	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn 𝐶) ↔ (𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
139	99, 138	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ ({𝑚} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑚} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀))))
140	5	cvmsss 32627	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (𝑆‘𝑀) → 𝑇 ⊆ 𝐶)
141	31, 140	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ 𝐶)
142	45	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ 𝑇)
143	141, 142	sseldd 3916	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
144		elssuni 4830	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇)
145	142, 144	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ⊆ ∪ 𝑇)
146	5	cvmsuni 32629	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (𝑆‘𝑀) → ∪ 𝑇 = (^◡𝐹 “ 𝑀))
147	31, 146	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑇 = (^◡𝐹 “ 𝑀))
148	145, 147	sseqtrd 3955	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ (^◡𝐹 “ 𝑀))
149	5	cvmsrcl 32624	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ (𝑆‘𝑀) → 𝑀 ∈ 𝐽)
150	31, 149	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ 𝐽)
151		cnima 21870	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝑀 ∈ 𝐽) → (^◡𝐹 “ 𝑀) ∈ 𝐶)
152	130, 150, 151	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ 𝑀) ∈ 𝐶)
153		restopn2 21782	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑀) ∈ 𝐶) → (𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑀)) ↔ (𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ (^◡𝐹 “ 𝑀))))
154	101, 152, 153	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑀)) ↔ (𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ (^◡𝐹 “ 𝑀))))
155	143, 148, 154	mpbir2and 712	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑀)))
156	155	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑀)))
157	5	cvmscld 32633	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑀) ∧ 𝑊 ∈ 𝑇) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑀))))
158	6, 31, 142, 157	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑀))))
159	158	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑀))))
160		cvmlift2lem9.10	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
161	160	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑍 ∈ 𝑉)
162		opelxpi 5556	. . . . . . . . . 10 ⊢ ((𝑚 ∈ {𝑚} ∧ 𝑍 ∈ 𝑉) → ⟨𝑚, 𝑍⟩ ∈ ({𝑚} × 𝑉))
163	52, 161, 162	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ⟨𝑚, 𝑍⟩ ∈ ({𝑚} × 𝑉))
164	81, 84	sstrd 3925	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
165	4	restuni 21767	. . . . . . . . . 10 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑚} × 𝑉) ⊆ ((0[,]1) × (0[,]1))) → ({𝑚} × 𝑉) = ∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉)))
166	15, 164, 165	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ({𝑚} × 𝑉) = ∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉)))
167	163, 166	eleqtrd 2892	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ⟨𝑚, 𝑍⟩ ∈ ∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉)))
168		df-ov 7138	. . . . . . . . . 10 ⊢ (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑍) = ((𝐾 ↾ ({𝑚} × 𝑉))‘⟨𝑚, 𝑍⟩)
169		ovres 7294	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ {𝑚} ∧ 𝑍 ∈ 𝑉) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑍) = (𝑚𝐾𝑍))
170	52, 161, 169	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑍) = (𝑚𝐾𝑍))
171		snidg 4559	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑉 → 𝑍 ∈ {𝑍})
172	160, 171	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
173	172	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → 𝑍 ∈ {𝑍})
174		ovres 7294	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ 𝑈 ∧ 𝑍 ∈ {𝑍}) → (𝑚(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = (𝑚𝐾𝑍))
175	106, 173, 174	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = (𝑚𝐾𝑍))
176	170, 175	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑍) = (𝑚(𝐾 ↾ (𝑈 × {𝑍}))𝑍))
177	168, 176	syl5eqr 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × 𝑉))‘⟨𝑚, 𝑍⟩) = (𝑚(𝐾 ↾ (𝑈 × {𝑍}))𝑍))
178		eqid 2798	. . . . . . . . . . . . 13 ⊢ ∪ ((II ×_t II) ↾_t (𝑈 × {𝑍})) = ∪ ((II ×_t II) ↾_t (𝑈 × {𝑍}))
179	2	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → II ∈ Top)
180		snex 5297	. . . . . . . . . . . . . . . 16 ⊢ {𝑍} ∈ V
181	180	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑍} ∈ V)
182		txrest 22236	. . . . . . . . . . . . . . 15 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ (𝑈 ∈ II ∧ {𝑍} ∈ V)) → ((II ×_t II) ↾_t (𝑈 × {𝑍})) = ((II ↾_t 𝑈) ×_t (II ↾_t {𝑍})))
183	179, 179, 17, 181, 182	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((II ×_t II) ↾_t (𝑈 × {𝑍})) = ((II ↾_t 𝑈) ×_t (II ↾_t {𝑍})))
184		cvmlift2lem9.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (II ↾_t 𝑈) ∈ Conn)
185	26, 160	sseldd 3916	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑍 ∈ (0[,]1))
186		restsn2 21776	. . . . . . . . . . . . . . . . 17 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝑍 ∈ (0[,]1)) → (II ↾_t {𝑍}) = 𝒫 {𝑍})
187	63, 185, 186	sylancr 590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (II ↾_t {𝑍}) = 𝒫 {𝑍})
188		pwsn 4792	. . . . . . . . . . . . . . . . 17 ⊢ 𝒫 {𝑍} = {∅, {𝑍}}
189		indisconn 22023	. . . . . . . . . . . . . . . . 17 ⊢ {∅, {𝑍}} ∈ Conn
190	188, 189	eqeltri 2886	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 {𝑍} ∈ Conn
191	187, 190	eqeltrdi 2898	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (II ↾_t {𝑍}) ∈ Conn)
192		txconn 22294	. . . . . . . . . . . . . . 15 ⊢ (((II ↾_t 𝑈) ∈ Conn ∧ (II ↾_t {𝑍}) ∈ Conn) → ((II ↾_t 𝑈) ×_t (II ↾_t {𝑍})) ∈ Conn)
193	184, 191, 192	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((II ↾_t 𝑈) ×_t (II ↾_t {𝑍})) ∈ Conn)
194	183, 193	eqeltrd 2890	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((II ×_t II) ↾_t (𝑈 × {𝑍})) ∈ Conn)
195		cvmlift2lem9.11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn 𝐶))
196	101, 103	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
197		df-ima 5532	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 “ (𝑈 × {𝑍})) = ran (𝐾 ↾ (𝑈 × {𝑍}))
198	160	snssd 4702	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {𝑍} ⊆ 𝑉)
199		xpss2 5539	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑍} ⊆ 𝑉 → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
200		imass2 5932	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉) → (𝐾 “ (𝑈 × {𝑍})) ⊆ (𝐾 “ (𝑈 × 𝑉)))
201	198, 199, 200	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 “ (𝑈 × {𝑍})) ⊆ (𝐾 “ (𝑈 × 𝑉)))
202	201, 124	sstrd 3925	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾 “ (𝑈 × {𝑍})) ⊆ (^◡𝐹 “ 𝑀))
203	197, 202	eqsstrrid 3964	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran (𝐾 ↾ (𝑈 × {𝑍})) ⊆ (^◡𝐹 “ 𝑀))
204		cnrest2 21891	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran (𝐾 ↾ (𝑈 × {𝑍})) ⊆ (^◡𝐹 “ 𝑀) ∧ (^◡𝐹 “ 𝑀) ⊆ 𝐵) → ((𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn 𝐶) ↔ (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
205	196, 203, 135, 204	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn 𝐶) ↔ (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
206	195, 205	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑍})) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀))))
207		opelxpi 5556	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ {𝑍}) → ⟨𝑋, 𝑍⟩ ∈ (𝑈 × {𝑍}))
208	21, 172, 207	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝑈 × {𝑍}))
209	185	snssd 4702	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑍} ⊆ (0[,]1))
210		xpss12 5534	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ⊆ (0[,]1) ∧ {𝑍} ⊆ (0[,]1)) → (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1)))
211	20, 209, 210	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1)))
212	4	restuni 21767	. . . . . . . . . . . . . . 15 ⊢ (((II ×_t II) ∈ Top ∧ (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1))) → (𝑈 × {𝑍}) = ∪ ((II ×_t II) ↾_t (𝑈 × {𝑍})))
213	15, 211, 212	sylancr 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 × {𝑍}) = ∪ ((II ×_t II) ↾_t (𝑈 × {𝑍})))
214	208, 213	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ ∪ ((II ×_t II) ↾_t (𝑈 × {𝑍})))
215		df-ov 7138	. . . . . . . . . . . . . . 15 ⊢ (𝑋(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = ((𝐾 ↾ (𝑈 × {𝑍}))‘⟨𝑋, 𝑍⟩)
216		ovres 7294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑍 ∈ {𝑍}) → (𝑋(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = (𝑋𝐾𝑍))
217	21, 172, 216	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = (𝑋𝐾𝑍))
218		snidg 4559	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ {𝑋})
219	21, 218	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
220		ovres 7294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑍 ∈ 𝑉) → (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑍) = (𝑋𝐾𝑍))
221	219, 160, 220	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑍) = (𝑋𝐾𝑍))
222	217, 221	eqtr4d 2836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋(𝐾 ↾ (𝑈 × {𝑍}))𝑍) = (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑍))
223	215, 222	syl5eqr 2847	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 ↾ (𝑈 × {𝑍}))‘⟨𝑋, 𝑍⟩) = (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑍))
224		eqid 2798	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉)) = ∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉))
225		snex 5297	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑋} ∈ V
226	225	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑋} ∈ V)
227		txrest 22236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ ({𝑋} ∈ V ∧ 𝑉 ∈ II)) → ((II ×_t II) ↾_t ({𝑋} × 𝑉)) = ((II ↾_t {𝑋}) ×_t (II ↾_t 𝑉)))
228	179, 179, 226, 23, 227	syl22anc 837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((II ×_t II) ↾_t ({𝑋} × 𝑉)) = ((II ↾_t {𝑋}) ×_t (II ↾_t 𝑉)))
229		restsn2 21776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝑋 ∈ (0[,]1)) → (II ↾_t {𝑋}) = 𝒫 {𝑋})
230	63, 22, 229	sylancr 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (II ↾_t {𝑋}) = 𝒫 {𝑋})
231		pwsn 4792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 {𝑋} = {∅, {𝑋}}
232		indisconn 22023	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅, {𝑋}} ∈ Conn
233	231, 232	eqeltri 2886	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝒫 {𝑋} ∈ Conn
234	230, 233	eqeltrdi 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (II ↾_t {𝑋}) ∈ Conn)
235		txconn 22294	. . . . . . . . . . . . . . . . . . 19 ⊢ (((II ↾_t {𝑋}) ∈ Conn ∧ (II ↾_t 𝑉) ∈ Conn) → ((II ↾_t {𝑋}) ×_t (II ↾_t 𝑉)) ∈ Conn)
236	234, 72, 235	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((II ↾_t {𝑋}) ×_t (II ↾_t 𝑉)) ∈ Conn)
237	228, 236	eqeltrd 2890	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((II ×_t II) ↾_t ({𝑋} × 𝑉)) ∈ Conn)
238	1, 6, 7, 8, 9, 10, 11	cvmlift2lem6 32668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑋} × (0[,]1))) Cn 𝐶))
239	22, 238	mpdan 686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑋} × (0[,]1))) Cn 𝐶))
240		xpss2 5539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 ⊆ (0[,]1) → ({𝑋} × 𝑉) ⊆ ({𝑋} × (0[,]1)))
241	23, 25, 240	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ({𝑋} × 𝑉) ⊆ ({𝑋} × (0[,]1)))
242	22	snssd 4702	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {𝑋} ⊆ (0[,]1))
243		xpss1 5538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({𝑋} ⊆ (0[,]1) → ({𝑋} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1)))
244	242, 243	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ({𝑋} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1)))
245	4	restuni 21767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑋} × (0[,]1)) ⊆ ((0[,]1) × (0[,]1))) → ({𝑋} × (0[,]1)) = ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1))))
246	15, 244, 245	sylancr 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ({𝑋} × (0[,]1)) = ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1))))
247	241, 246	sseqtrd 3955	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ({𝑋} × 𝑉) ⊆ ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1))))
248		eqid 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1))) = ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1)))
249	248	cnrest 21890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×_t II) ↾_t ({𝑋} × (0[,]1))) Cn 𝐶) ∧ ({𝑋} × 𝑉) ⊆ ∪ ((II ×_t II) ↾_t ({𝑋} × (0[,]1)))) → ((𝐾 ↾ ({𝑋} × (0[,]1))) ↾ ({𝑋} × 𝑉)) ∈ ((((II ×_t II) ↾_t ({𝑋} × (0[,]1))) ↾_t ({𝑋} × 𝑉)) Cn 𝐶))
250	239, 247, 249	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × (0[,]1))) ↾ ({𝑋} × 𝑉)) ∈ ((((II ×_t II) ↾_t ({𝑋} × (0[,]1))) ↾_t ({𝑋} × 𝑉)) Cn 𝐶))
251	241	resabs1d 5849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × (0[,]1))) ↾ ({𝑋} × 𝑉)) = (𝐾 ↾ ({𝑋} × 𝑉)))
252	225, 93	xpex 7456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑋} × (0[,]1)) ∈ V
253	252	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ({𝑋} × (0[,]1)) ∈ V)
254		restabs 21770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑋} × 𝑉) ⊆ ({𝑋} × (0[,]1)) ∧ ({𝑋} × (0[,]1)) ∈ V) → (((II ×_t II) ↾_t ({𝑋} × (0[,]1))) ↾_t ({𝑋} × 𝑉)) = ((II ×_t II) ↾_t ({𝑋} × 𝑉)))
255	16, 241, 253, 254	syl3anc 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((II ×_t II) ↾_t ({𝑋} × (0[,]1))) ↾_t ({𝑋} × 𝑉)) = ((II ×_t II) ↾_t ({𝑋} × 𝑉)))
256	255	oveq1d 7150	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((II ×_t II) ↾_t ({𝑋} × (0[,]1))) ↾_t ({𝑋} × 𝑉)) Cn 𝐶) = (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn 𝐶))
257	250, 251, 256	3eltr3d 2904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn 𝐶))
258		df-ima 5532	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 “ ({𝑋} × 𝑉)) = ran (𝐾 ↾ ({𝑋} × 𝑉))
259	21	snssd 4702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑋} ⊆ 𝑈)
260		xpss1 5538	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑋} ⊆ 𝑈 → ({𝑋} × 𝑉) ⊆ (𝑈 × 𝑉))
261		imass2 5932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({𝑋} × 𝑉) ⊆ (𝑈 × 𝑉) → (𝐾 “ ({𝑋} × 𝑉)) ⊆ (𝐾 “ (𝑈 × 𝑉)))
262	259, 260, 261	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐾 “ ({𝑋} × 𝑉)) ⊆ (𝐾 “ (𝑈 × 𝑉)))
263	262, 124	sstrd 3925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐾 “ ({𝑋} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
264	258, 263	eqsstrrid 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran (𝐾 ↾ ({𝑋} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀))
265		cnrest2 21891	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran (𝐾 ↾ ({𝑋} × 𝑉)) ⊆ (^◡𝐹 “ 𝑀) ∧ (^◡𝐹 “ 𝑀) ⊆ 𝐵) → ((𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn 𝐶) ↔ (𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
266	196, 264, 135, 265	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn 𝐶) ↔ (𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀)))))
267	257, 266	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 ↾ ({𝑋} × 𝑉)) ∈ (((II ×_t II) ↾_t ({𝑋} × 𝑉)) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑀))))
268		opelxpi 5556	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝑉))
269	219, 27, 268	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝑉))
270	259, 260	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ({𝑋} × 𝑉) ⊆ (𝑈 × 𝑉))
271	270, 50	sstrd 3925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ({𝑋} × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
272	4	restuni 21767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((II ×_t II) ∈ Top ∧ ({𝑋} × 𝑉) ⊆ ((0[,]1) × (0[,]1))) → ({𝑋} × 𝑉) = ∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉)))
273	15, 271, 272	sylancr 590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ({𝑋} × 𝑉) = ∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉)))
274	269, 273	eleqtrd 2892	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉)))
275		df-ov 7138	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑌) = ((𝐾 ↾ ({𝑋} × 𝑉))‘⟨𝑋, 𝑌⟩)
276		ovres 7294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑌 ∈ 𝑉) → (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑌) = (𝑋𝐾𝑌))
277	219, 27, 276	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑌) = (𝑋𝐾𝑌))
278	275, 277	syl5eqr 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × 𝑉))‘⟨𝑋, 𝑌⟩) = (𝑋𝐾𝑌))
279	45	simprd 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋𝐾𝑌) ∈ 𝑊)
280	278, 279	eqeltrd 2890	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × 𝑉))‘⟨𝑋, 𝑌⟩) ∈ 𝑊)
281	224, 237, 267, 155, 158, 274, 280	conncn 22031	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾 ↾ ({𝑋} × 𝑉)):∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉))⟶𝑊)
282	273	feq2d 6473	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾 ↾ ({𝑋} × 𝑉)):({𝑋} × 𝑉)⟶𝑊 ↔ (𝐾 ↾ ({𝑋} × 𝑉)):∪ ((II ×_t II) ↾_t ({𝑋} × 𝑉))⟶𝑊))
283	281, 282	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 ↾ ({𝑋} × 𝑉)):({𝑋} × 𝑉)⟶𝑊)
284	283, 219, 160	fovrnd 7300	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋(𝐾 ↾ ({𝑋} × 𝑉))𝑍) ∈ 𝑊)
285	223, 284	eqeltrd 2890	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾 ↾ (𝑈 × {𝑍}))‘⟨𝑋, 𝑍⟩) ∈ 𝑊)
286	178, 194, 206, 155, 158, 214, 285	conncn 22031	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})):∪ ((II ×_t II) ↾_t (𝑈 × {𝑍}))⟶𝑊)
287	213	feq2d 6473	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↾ (𝑈 × {𝑍})):(𝑈 × {𝑍})⟶𝑊 ↔ (𝐾 ↾ (𝑈 × {𝑍})):∪ ((II ×_t II) ↾_t (𝑈 × {𝑍}))⟶𝑊))
288	286, 287	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})):(𝑈 × {𝑍})⟶𝑊)
289	288	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ (𝑈 × {𝑍})):(𝑈 × {𝑍})⟶𝑊)
290	289, 106, 173	fovrnd 7300	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ (𝑈 × {𝑍}))𝑍) ∈ 𝑊)
291	177, 290	eqeltrd 2890	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × 𝑉))‘⟨𝑚, 𝑍⟩) ∈ 𝑊)
292	56, 76, 139, 156, 159, 167, 291	conncn 22031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ ({𝑚} × 𝑉)):∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉))⟶𝑊)
293	166	feq2d 6473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → ((𝐾 ↾ ({𝑚} × 𝑉)):({𝑚} × 𝑉)⟶𝑊 ↔ (𝐾 ↾ ({𝑚} × 𝑉)):∪ ((II ×_t II) ↾_t ({𝑚} × 𝑉))⟶𝑊))
294	292, 293	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝐾 ↾ ({𝑚} × 𝑉)):({𝑚} × 𝑉)⟶𝑊)
295	294, 52, 53	fovrnd 7300	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚(𝐾 ↾ ({𝑚} × 𝑉))𝑛) ∈ 𝑊)
296	55, 295	eqeltrrd 2891	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉)) → (𝑚𝐾𝑛) ∈ 𝑊)
297	296	ralrimivva 3156	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑈 ∀𝑛 ∈ 𝑉 (𝑚𝐾𝑛) ∈ 𝑊)
298		funimassov 7305	. . . 4 ⊢ ((Fun 𝐾 ∧ (𝑈 × 𝑉) ⊆ dom 𝐾) → ((𝐾 “ (𝑈 × 𝑉)) ⊆ 𝑊 ↔ ∀𝑚 ∈ 𝑈 ∀𝑛 ∈ 𝑉 (𝑚𝐾𝑛) ∈ 𝑊))
299	119, 121, 298	syl2anc 587	. . 3 ⊢ (𝜑 → ((𝐾 “ (𝑈 × 𝑉)) ⊆ 𝑊 ↔ ∀𝑚 ∈ 𝑈 ∀𝑛 ∈ 𝑉 (𝑚𝐾𝑛) ∈ 𝑊))
300	297, 299	mpbird 260	. 2 ⊢ (𝜑 → (𝐾 “ (𝑈 × 𝑉)) ⊆ 𝑊)
301	1, 4, 5, 6, 12, 14, 16, 30, 31, 48, 50, 300	cvmlift2lem9a 32663	1 ⊢ (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 {cpr 4527 ⟨cop 4531 ∪ cuni 4800 ↦ cmpt 5110 × cxp 5517 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 1c1 10527 [,]cicc 12729 ↾_t crest 16686 Topctop 21498 TopOnctopon 21515 Clsdccld 21621 Cn ccn 21829 Conncconn 22016 ×_t ctx 22165 Homeochmeo 22358 IIcii 23480 CovMap ccvm 32615

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-cmp 21992 df-conn 22017 df-lly 22071 df-nlly 22072 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-ii 23482 df-htpy 23575 df-phtpy 23576 df-phtpc 23597 df-pconn 32581 df-sconn 32582 df-cvm 32616

This theorem is referenced by: cvmlift2lem10 32672

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