cvmlift2lem11 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem11 33275

Description: Lemma for cvmlift2 33278. (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))
cvmlift2.m	⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)}
cvmlift2lem11.1	⊢ (𝜑 → 𝑈 ∈ II)
cvmlift2lem11.2	⊢ (𝜑 → 𝑉 ∈ II)
cvmlift2lem11.3	⊢ (𝜑 → 𝑌 ∈ 𝑉)
cvmlift2lem11.4	⊢ (𝜑 → 𝑍 ∈ 𝑉)
cvmlift2lem11.5	⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶)))

**Assertion**
Ref	Expression
cvmlift2lem11	⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀))

Distinct variable groups: 𝑤,𝑓,𝑥,𝑦,𝑧,𝐹 𝜑,𝑓,𝑤,𝑥,𝑦,𝑧 𝑥,𝑀,𝑦,𝑧 𝑓,𝐽,𝑤,𝑥,𝑦,𝑧 𝑤,𝑈,𝑧 𝑓,𝐺,𝑤,𝑥,𝑦,𝑧 𝑤,𝑉 𝑓,𝐻,𝑤,𝑥,𝑦,𝑧 𝑧,𝑍 𝐶,𝑓,𝑤,𝑥,𝑦,𝑧 𝑃,𝑓,𝑥,𝑦,𝑧 𝑤,𝐵,𝑥,𝑦,𝑧 𝑓,𝑌,𝑤,𝑥,𝑦,𝑧 𝑓,𝐾,𝑤,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐵(𝑓) 𝑃(𝑤) 𝑈(𝑥,𝑦,𝑓) 𝑀(𝑤,𝑓) 𝑉(𝑥,𝑦,𝑧,𝑓) 𝑍(𝑥,𝑦,𝑤,𝑓)

**Proof of Theorem cvmlift2lem11**
Step	Hyp	Ref	Expression
1		cvmlift2lem11.1	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ II)
2	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑈 ∈ II)
3		elssuni 4871	. . . . . . 7 ⊢ (𝑈 ∈ II → 𝑈 ⊆ ∪ II)
4		iiuni 24044	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 4	sseqtrrdi 3972	. . . . . 6 ⊢ (𝑈 ∈ II → 𝑈 ⊆ (0[,]1))
6	2, 5	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑈 ⊆ (0[,]1))
7		cvmlift2lem11.4	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
8		cvmlift2lem11.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ II)
9		elunii 4844	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ ∪ II)
10	9, 4	eleqtrrdi 2850	. . . . . . . 8 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ (0[,]1))
11	7, 8, 10	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (0[,]1))
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑍 ∈ (0[,]1))
13	12	snssd 4742	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ (0[,]1))
14		xpss12 5604	. . . . 5 ⊢ ((𝑈 ⊆ (0[,]1) ∧ {𝑍} ⊆ (0[,]1)) → (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1)))
15	6, 13, 14	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1)))
16		cvmlift2lem11.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
17	16	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑌 ∈ 𝑉)
18		cvmlift2.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ∪ 𝐶
19		cvmlift2.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
20		cvmlift2.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
21		cvmlift2.p	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
22		cvmlift2.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
23		cvmlift2.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
24		cvmlift2.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))
25	18, 19, 20, 21, 22, 23, 24	cvmlift2lem5 33269	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
27	8	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ∈ II)
28		elssuni 4871	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ II → 𝑉 ⊆ ∪ II)
29	28, 4	sseqtrrdi 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ II → 𝑉 ⊆ (0[,]1))
30	27, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ⊆ (0[,]1))
31	30, 17	sseldd 3922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑌 ∈ (0[,]1))
32	31	snssd 4742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑌} ⊆ (0[,]1))
33		xpss12 5604	. . . . . . . . . . . 12 ⊢ ((𝑈 ⊆ (0[,]1) ∧ {𝑌} ⊆ (0[,]1)) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
34	6, 32, 33	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
35	26, 34	fssresd 6641	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × {𝑌})):(𝑈 × {𝑌})⟶𝐵)
36	34	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
37		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝑧 ∈ (𝑈 × {𝑌}))
38		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ 𝑀)
39		cvmlift2.m	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)}
40	38, 39	sseqtrdi 3971	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
41		ssrab 4006	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)} ↔ ((𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)) ∧ ∀𝑧 ∈ (𝑈 × {𝑌})𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)))
42	41	simprbi 497	. . . . . . . . . . . . . 14 ⊢ ((𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)} → ∀𝑧 ∈ (𝑈 × {𝑌})𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
43	40, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∀𝑧 ∈ (𝑈 × {𝑌})𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
44	43	r19.21bi 3134	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
45		iitopon 24042	. . . . . . . . . . . . . . 15 ⊢ II ∈ (TopOn‘(0[,]1))
46		txtopon 22742	. . . . . . . . . . . . . . 15 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
47	45, 45, 46	mp2an 689	. . . . . . . . . . . . . 14 ⊢ (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
48	47	toponunii 22065	. . . . . . . . . . . . 13 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
49	48	cnpresti 22439	. . . . . . . . . . . 12 ⊢ (((𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)) ∧ 𝑧 ∈ (𝑈 × {𝑌}) ∧ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
50	36, 37, 44, 49	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
51	50	ralrimiva 3103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
52		resttopon 22312	. . . . . . . . . . . 12 ⊢ (((II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1))) ∧ (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1))) → ((II ×_t II) ↾_t (𝑈 × {𝑌})) ∈ (TopOn‘(𝑈 × {𝑌})))
53	47, 34, 52	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ((II ×_t II) ↾_t (𝑈 × {𝑌})) ∈ (TopOn‘(𝑈 × {𝑌})))
54		cvmtop1 33222	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
55	19, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Top)
56	55	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ Top)
57	18	toptopon 22066	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
58	56, 57	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ (TopOn‘𝐵))
59		cncnp 22431	. . . . . . . . . . 11 ⊢ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) ∈ (TopOn‘(𝑈 × {𝑌})) ∧ 𝐶 ∈ (TopOn‘𝐵)) → ((𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶) ↔ ((𝐾 ↾ (𝑈 × {𝑌})):(𝑈 × {𝑌})⟶𝐵 ∧ ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))))
60	53, 58, 59	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ((𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶) ↔ ((𝐾 ↾ (𝑈 × {𝑌})):(𝑈 × {𝑌})⟶𝐵 ∧ ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))))
61	35, 51, 60	mpbir2and 710	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
62		sneq 4571	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑌 → {𝑤} = {𝑌})
63	62	xpeq2d 5619	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → (𝑈 × {𝑤}) = (𝑈 × {𝑌}))
64	63	reseq2d 5891	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (𝐾 ↾ (𝑈 × {𝑤})) = (𝐾 ↾ (𝑈 × {𝑌})))
65	63	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → ((II ×_t II) ↾_t (𝑈 × {𝑤})) = ((II ×_t II) ↾_t (𝑈 × {𝑌})))
66	65	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) = (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
67	64, 66	eleq12d 2833	. . . . . . . . . 10 ⊢ (𝑤 = 𝑌 → ((𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) ↔ (𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶)))
68	67	rspcev 3561	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶)) → ∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶))
69	17, 61, 68	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶))
70		cvmlift2lem11.5	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶)))
71	70	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶)) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))
72	69, 71	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))
73	72	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))
74	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑍 ∈ 𝑉)
75	74	snssd 4742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ 𝑉)
76		xpss2 5609	. . . . . . . . 9 ⊢ ({𝑍} ⊆ 𝑉 → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
77	75, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
78		iitop 24043	. . . . . . . . . 10 ⊢ II ∈ Top
79	78, 78	txtopi 22741	. . . . . . . . 9 ⊢ (II ×_t II) ∈ Top
80		xpss12 5604	. . . . . . . . . 10 ⊢ ((𝑈 ⊆ (0[,]1) ∧ 𝑉 ⊆ (0[,]1)) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
81	6, 30, 80	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
82	48	restuni 22313	. . . . . . . . 9 ⊢ (((II ×_t II) ∈ Top ∧ (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1))) → (𝑈 × 𝑉) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
83	79, 81, 82	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
84	77, 83	sseqtrd 3961	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
85	84	sselda 3921	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
86		eqid 2738	. . . . . . 7 ⊢ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉))
87	86	cncnpi 22429	. . . . . 6 ⊢ (((𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶) ∧ 𝑧 ∈ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉))) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ ((((II ×_t II) ↾_t (𝑈 × 𝑉)) CnP 𝐶)‘𝑧))
88	73, 85, 87	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ ((((II ×_t II) ↾_t (𝑈 × 𝑉)) CnP 𝐶)‘𝑧))
89	79	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (II ×_t II) ∈ Top)
90	81	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
91	78	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → II ∈ Top)
92		txopn 22753	. . . . . . . . . 10 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ (𝑈 ∈ II ∧ 𝑉 ∈ II)) → (𝑈 × 𝑉) ∈ (II ×_t II))
93	91, 91, 2, 27, 92	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) ∈ (II ×_t II))
94		isopn3i 22233	. . . . . . . . 9 ⊢ (((II ×_t II) ∈ Top ∧ (𝑈 × 𝑉) ∈ (II ×_t II)) → ((int‘(II ×_t II))‘(𝑈 × 𝑉)) = (𝑈 × 𝑉))
95	79, 93, 94	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ((int‘(II ×_t II))‘(𝑈 × 𝑉)) = (𝑈 × 𝑉))
96	77, 95	sseqtrrd 3962	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
97	96	sselda 3921	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
98	25	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
99	48, 18	cnprest 22440	. . . . . 6 ⊢ ((((II ×_t II) ∈ Top ∧ (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1))) ∧ (𝑧 ∈ ((int‘(II ×_t II))‘(𝑈 × 𝑉)) ∧ 𝐾:((0[,]1) × (0[,]1))⟶𝐵)) → (𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧) ↔ (𝐾 ↾ (𝑈 × 𝑉)) ∈ ((((II ×_t II) ↾_t (𝑈 × 𝑉)) CnP 𝐶)‘𝑧)))
100	89, 90, 97, 98, 99	syl22anc 836	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧) ↔ (𝐾 ↾ (𝑈 × 𝑉)) ∈ ((((II ×_t II) ↾_t (𝑈 × 𝑉)) CnP 𝐶)‘𝑧)))
101	88, 100	mpbird 256	. . . 4 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
102	15, 101	ssrabdv 4007	. . 3 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
103	102, 39	sseqtrrdi 3972	. 2 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ 𝑀)
104	103	ex 413	1 ⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 ⊆ wss 3887 {csn 4561 ∪ cuni 4839 ↦ cmpt 5157 × cxp 5587 ↾ cres 5591 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 ∈ cmpo 7277 0cc0 10871 1c1 10872 [,]cicc 13082 ↾_t crest 17131 Topctop 22042 TopOnctopon 22059 intcnt 22168 Cn ccn 22375 CnP ccnp 22376 ×_t ctx 22711 IIcii 24038 CovMap ccvm 33217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-ec 8500 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-cmp 22538 df-conn 22563 df-lly 22617 df-nlly 22618 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-ii 24040 df-htpy 24133 df-phtpy 24134 df-phtpc 24155 df-pconn 33183 df-sconn 33184 df-cvm 33218

This theorem is referenced by: cvmlift2lem12 33276

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