cvmlift2lem11 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem11 35511

Description: Lemma for cvmlift2 35514. (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 4882	. . . . . . 7 ⊢ (𝑈 ∈ II → 𝑈 ⊆ ∪ II)
4		iiuni 24858	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 4	sseqtrrdi 3964	. . . . . 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 4856	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ ∪ II)
10	9, 4	eleqtrrdi 2848	. . . . . . . 8 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑉 ∈ II) → 𝑍 ∈ (0[,]1))
11	7, 8, 10	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (0[,]1))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑍 ∈ (0[,]1))
13	12	snssd 4753	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ (0[,]1))
14		xpss12 5639	. . . . 5 ⊢ ((𝑈 ⊆ (0[,]1) ∧ {𝑍} ⊆ (0[,]1)) → (𝑈 × {𝑍}) ⊆ ((0[,]1) × (0[,]1)))
15	6, 13, 14	syl2anc 585	. . . 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 35505	. . . . . . . . . . . 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 4882	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ II → 𝑉 ⊆ ∪ II)
29	28, 4	sseqtrrdi 3964	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ II → 𝑉 ⊆ (0[,]1))
30	27, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑉 ⊆ (0[,]1))
31	30, 17	sseldd 3923	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝑌 ∈ (0[,]1))
32	31	snssd 4753	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑌} ⊆ (0[,]1))
33		xpss12 5639	. . . . . . . . . . . 12 ⊢ ((𝑈 ⊆ (0[,]1) ∧ {𝑌} ⊆ (0[,]1)) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
34	6, 32, 33	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ ((0[,]1) × (0[,]1)))
35	26, 34	fssresd 6701	. . . . . . . . . 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 3963	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑌}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
41		ssrab 4012	. . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧))
45		iitopon 24856	. . . . . . . . . . . . . . 15 ⊢ II ∈ (TopOn‘(0[,]1))
46		txtopon 23566	. . . . . . . . . . . . . . 15 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
47	45, 45, 46	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (II ×_t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
48	47	toponunii 22891	. . . . . . . . . . . . 13 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
49	48	cnpresti 23263	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑌})) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
51	50	ralrimiva 3130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ∀𝑧 ∈ (𝑈 × {𝑌})(𝐾 ↾ (𝑈 × {𝑌})) ∈ ((((II ×_t II) ↾_t (𝑈 × {𝑌})) CnP 𝐶)‘𝑧))
52		resttopon 23136	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ((II ×_t II) ↾_t (𝑈 × {𝑌})) ∈ (TopOn‘(𝑈 × {𝑌})))
54		cvmtop1 35458	. . . . . . . . . . . . . 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 22892	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
58	56, 57	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → 𝐶 ∈ (TopOn‘𝐵))
59		cncnp 23255	. . . . . . . . . . 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 585	. . . . . . . . . 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 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝐾 ↾ (𝑈 × {𝑌})) ∈ (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
62		sneq 4578	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑌 → {𝑤} = {𝑌})
63	62	xpeq2d 5654	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → (𝑈 × {𝑤}) = (𝑈 × {𝑌}))
64	63	reseq2d 5938	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (𝐾 ↾ (𝑈 × {𝑤})) = (𝐾 ↾ (𝑈 × {𝑌})))
65	63	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑌 → ((II ×_t II) ↾_t (𝑈 × {𝑤})) = ((II ×_t II) ↾_t (𝑈 × {𝑌})))
66	65	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑌 → (((II ×_t II) ↾_t (𝑈 × {𝑤})) Cn 𝐶) = (((II ×_t II) ↾_t (𝑈 × {𝑌})) Cn 𝐶))
67	64, 66	eleq12d 2831	. . . . . . . . . 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 3565	. . . . . . . . 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 585	. . . . . . . 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 592	. . . . . . 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 4753	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → {𝑍} ⊆ 𝑉)
76		xpss2 5644	. . . . . . . . 9 ⊢ ({𝑍} ⊆ 𝑉 → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
77	75, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ (𝑈 × 𝑉))
78		iitop 24857	. . . . . . . . . 10 ⊢ II ∈ Top
79	78, 78	txtopi 23565	. . . . . . . . 9 ⊢ (II ×_t II) ∈ Top
80		xpss12 5639	. . . . . . . . . 10 ⊢ ((𝑈 ⊆ (0[,]1) ∧ 𝑉 ⊆ (0[,]1)) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
81	6, 30, 80	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1)))
82	48	restuni 23137	. . . . . . . . 9 ⊢ (((II ×_t II) ∈ Top ∧ (𝑈 × 𝑉) ⊆ ((0[,]1) × (0[,]1))) → (𝑈 × 𝑉) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
83	79, 81, 82	sylancr 588	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
84	77, 83	sseqtrd 3959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
85	84	sselda 3922	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)))
86		eqid 2737	. . . . . . 7 ⊢ ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉)) = ∪ ((II ×_t II) ↾_t (𝑈 × 𝑉))
87	86	cncnpi 23253	. . . . . 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 585	. . . . 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 23577	. . . . . . . . . 10 ⊢ (((II ∈ Top ∧ II ∈ Top) ∧ (𝑈 ∈ II ∧ 𝑉 ∈ II)) → (𝑈 × 𝑉) ∈ (II ×_t II))
93	91, 91, 2, 27, 92	syl22anc 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × 𝑉) ∈ (II ×_t II))
94		isopn3i 23057	. . . . . . . . 9 ⊢ (((II ×_t II) ∈ Top ∧ (𝑈 × 𝑉) ∈ (II ×_t II)) → ((int‘(II ×_t II))‘(𝑈 × 𝑉)) = (𝑈 × 𝑉))
95	79, 93, 94	sylancr 588	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → ((int‘(II ×_t II))‘(𝑈 × 𝑉)) = (𝑈 × 𝑉))
96	77, 95	sseqtrrd 3960	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
97	96	sselda 3922	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝑧 ∈ ((int‘(II ×_t II))‘(𝑈 × 𝑉)))
98	25	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) ∧ 𝑧 ∈ (𝑈 × {𝑍})) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
99	48, 18	cnprest 23264	. . . . . 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 839	. . . . 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 4014	. . 3 ⊢ ((𝜑 ∧ (𝑈 × {𝑌}) ⊆ 𝑀) → (𝑈 × {𝑍}) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)})
103	102, 39	sseqtrrdi 3964	. 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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3390 ⊆ wss 3890 {csn 4568 ∪ cuni 4851 ↦ cmpt 5167 × cxp 5622 ↾ cres 5626 ∘ ccom 5628 ⟶wf 6488 ‘cfv 6492 ℩crio 7316 (class class class)co 7360 ∈ cmpo 7362 0cc0 11029 1c1 11030 [,]cicc 13292 ↾_t crest 17374 Topctop 22868 TopOnctopon 22885 intcnt 22992 Cn ccn 23199 CnP ccnp 23200 ×_t ctx 23535 IIcii 24852 CovMap ccvm 35453

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-ec 8638 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-cn 23202 df-cnp 23203 df-cmp 23362 df-conn 23387 df-lly 23441 df-nlly 23442 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297 df-ii 24854 df-cncf 24855 df-htpy 24947 df-phtpy 24948 df-phtpc 24969 df-pconn 35419 df-sconn 35420 df-cvm 35454

This theorem is referenced by: cvmlift2lem12 35512

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