cvmlift2lem11 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem11 35307

Description: Lemma for cvmlift2 35310. (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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑈 ∈ II)
3		elssuni 4904	. . . . . . 7 ⊢ (𝑈 ∈ II → 𝑈 ⊆ ∪ II)
4		iiuni 24781	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 4	sseqtrrdi 3991	. . . . . 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 4879	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ ∪ II)
10	9, 4	eleqtrrdi 2840	. . . . . . . 8 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ (0[,]1))
11	7, 8, 10	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (0[,]1))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑍 ∈ (0[,]1))
13	12	snssd 4776	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ (0[,]1))
14		xpss12 5656	. . . . 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 480	. . . . . . . . 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 35301	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
27	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ∈ II)
28		elssuni 4904	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ II → 𝑉 ⊆ ∪ II)
29	28, 4	sseqtrrdi 3991	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ II → 𝑉 ⊆ (0[,]1))
30	27, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ⊆ (0[,]1))
31	30, 17	sseldd 3950	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑌 ∈ (0[,]1))
32	31	snssd 4776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑌} ⊆ (0[,]1))
33		xpss12 5656	. . . . . . . . . . . 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 6730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × {𝑌})):(𝑈 × {𝑌})⟶𝐵)
36	34	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
37		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝑧 ∈ (𝑈 × {𝑌}))
38		simpr 484	. . . . . . . . . . . . . . 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 3990	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
41		ssrab 4039	. . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 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 3230	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
45		iitopon 24779	. . . . . . . . . . . . . . 15 ⊢ II ∈ (TopOn‘(0[,]1))
46		txtopon 23485	. . . . . . . . . . . . . . 15 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
47	45, 45, 46	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
48	47	toponunii 22810	. . . . . . . . . . . . 13 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
49	48	cnpresti 23182	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
51	50	ralrimiva 3126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
52		resttopon 23055	. . . . . . . . . . . 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 35254	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
55	19, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Top)
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ Top)
57	18	toptopon 22811	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
58	56, 57	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ (TopOn‘𝐵))
59		cncnp 23174	. . . . . . . . . . 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 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
62		sneq 4602	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑌 → {𝑤} = {𝑌})
63	62	xpeq2d 5671	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → (𝑈 × {𝑤}) = (𝑈 × {𝑌}))
64	63	reseq2d 5953	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (𝐾 ↾ (𝑈 × {𝑤})) = (𝐾 ↾ (𝑈 × {𝑌})))
65	63	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → ((II ×_t II) ↾_t (𝑈 × {𝑤})) = ((II ×_t II) ↾_t (𝑈 × {𝑌})))
66	65	oveq1d 7405	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) = (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
67	64, 66	eleq12d 2823	. . . . . . . . . 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 3591	. . . . . . . . 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 406	. . . . . . . 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 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×_t II) ↾_t (𝑈 × 𝑉)) Cn 𝐶))
74	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑍 ∈ 𝑉)
75	74	snssd 4776	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ 𝑉)
76		xpss2 5661	. . . . . . . . 9 ⊢ ({𝑍} ⊆ 𝑉 → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
77	75, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
78		iitop 24780	. . . . . . . . . 10 ⊢ II ∈ Top
79	78, 78	txtopi 23484	. . . . . . . . 9 ⊢ (II ×_t II) ∈ Top
80		xpss12 5656	. . . . . . . . . 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 23056	. . . . . . . . 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 3986	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
85	84	sselda 3949	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
86		eqid 2730	. . . . . . 7 ⊢ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉))
87	86	cncnpi 23172	. . . . . 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 480	. . . . . 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 23496	. . . . . . . . . 10 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ (𝑈 ∈ II ∧ 𝑉 ∈ II)) → (𝑈 × 𝑉) ∈ (II ×_t II))
93	91, 91, 2, 27, 92	syl22anc 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) ∈ (II ×_t II))
94		isopn3i 22976	. . . . . . . . 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 3987	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
97	96	sselda 3949	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
98	25	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
99	48, 18	cnprest 23183	. . . . . 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 838	. . . . 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 257	. . . 4 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
102	15, 101	ssrabdv 4040	. . 3 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
103	102, 39	sseqtrrdi 3991	. 2 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ 𝑀)
104	103	ex 412	1 ⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 {crab 3408 ⊆ wss 3917 {csn 4592 ∪ cuni 4874 ↦ cmpt 5191 × cxp 5639 ↾ cres 5643 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 ℩crio 7346 (class class class)co 7390 ∈ cmpo 7392 0cc0 11075 1c1 11076 [,]cicc 13316 ↾_t crest 17390 Topctop 22787 TopOnctopon 22804 intcnt 22911 Cn ccn 23118 CnP ccnp 23119 ×_t ctx 23454 IIcii 24775 CovMap ccvm 35249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-ec 8676 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-cn 23121 df-cnp 23122 df-cmp 23281 df-conn 23306 df-lly 23360 df-nlly 23361 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-ii 24777 df-cncf 24778 df-htpy 24876 df-phtpy 24877 df-phtpc 24898 df-pconn 35215 df-sconn 35216 df-cvm 35250

This theorem is referenced by: cvmlift2lem12 35308

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