cvmlift2lem11 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem11 35285

Description: Lemma for cvmlift2 35288. (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 4891	. . . . . . 7 ⊢ (𝑈 ∈ II → 𝑈 ⊆ ∪ II)
4		iiuni 24790	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 4	sseqtrrdi 3979	. . . . . 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 4866	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ ∪ II)
10	9, 4	eleqtrrdi 2839	. . . . . . . 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 4763	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ (0[,]1))
14		xpss12 5638	. . . . 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 35279	. . . . . . . . . . . 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 4891	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ II → 𝑉 ⊆ ∪ II)
29	28, 4	sseqtrrdi 3979	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ II → 𝑉 ⊆ (0[,]1))
30	27, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ⊆ (0[,]1))
31	30, 17	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑌 ∈ (0[,]1))
32	31	snssd 4763	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑌} ⊆ (0[,]1))
33		xpss12 5638	. . . . . . . . . . . 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 6695	. . . . . . . . . 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 3978	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
41		ssrab 4026	. . . . . . . . . . . . . . 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 3221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
45		iitopon 24788	. . . . . . . . . . . . . . 15 ⊢ II ∈ (TopOn‘(0[,]1))
46		txtopon 23494	. . . . . . . . . . . . . . 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 22819	. . . . . . . . . . . . 13 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
49	48	cnpresti 23191	. . . . . . . . . . . 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 3121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
52		resttopon 23064	. . . . . . . . . . . 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 35232	. . . . . . . . . . . . . 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 22820	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
58	56, 57	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ (TopOn‘𝐵))
59		cncnp 23183	. . . . . . . . . . 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 4589	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑌 → {𝑤} = {𝑌})
63	62	xpeq2d 5653	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → (𝑈 × {𝑤}) = (𝑈 × {𝑌}))
64	63	reseq2d 5934	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (𝐾 ↾ (𝑈 × {𝑤})) = (𝐾 ↾ (𝑈 × {𝑌})))
65	63	oveq2d 7369	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → ((II ×_t II) ↾_t (𝑈 × {𝑤})) = ((II ×_t II) ↾_t (𝑈 × {𝑌})))
66	65	oveq1d 7368	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) = (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
67	64, 66	eleq12d 2822	. . . . . . . . . 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 3579	. . . . . . . . 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 4763	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ 𝑉)
76		xpss2 5643	. . . . . . . . 9 ⊢ ({𝑍} ⊆ 𝑉 → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
77	75, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
78		iitop 24789	. . . . . . . . . 10 ⊢ II ∈ Top
79	78, 78	txtopi 23493	. . . . . . . . 9 ⊢ (II ×_t II) ∈ Top
80		xpss12 5638	. . . . . . . . . 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 23065	. . . . . . . . 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 3974	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
85	84	sselda 3937	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
86		eqid 2729	. . . . . . 7 ⊢ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉))
87	86	cncnpi 23181	. . . . . 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 23505	. . . . . . . . . 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 22985	. . . . . . . . 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 3975	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
97	96	sselda 3937	. . . . . 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 23192	. . . . . 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 4027	. . 3 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
103	102, 39	sseqtrrdi 3979	. 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 3044 ∃wrex 3053 {crab 3396 ⊆ wss 3905 {csn 4579 ∪ cuni 4861 ↦ cmpt 5176 × cxp 5621 ↾ cres 5625 ∘ ccom 5627 ⟶wf 6482 ‘cfv 6486 ℩crio 7309 (class class class)co 7353 ∈ cmpo 7355 0cc0 11028 1c1 11029 [,]cicc 13269 ↾_t crest 17342 Topctop 22796 TopOnctopon 22813 intcnt 22920 Cn ccn 23127 CnP ccnp 23128 ×_t ctx 23463 IIcii 24784 CovMap ccvm 35227

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-ec 8634 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-cn 23130 df-cnp 23131 df-cmp 23290 df-conn 23315 df-lly 23369 df-nlly 23370 df-tx 23465 df-hmeo 23658 df-xms 24224 df-ms 24225 df-tms 24226 df-ii 24786 df-cncf 24787 df-htpy 24885 df-phtpy 24886 df-phtpc 24907 df-pconn 35193 df-sconn 35194 df-cvm 35228

This theorem is referenced by: cvmlift2lem12 35286

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