Theorem cvmlift2lem10 35506

Description: Lemma for cvmlift2 35510. (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))))})
cvmlift2lem10.1	⊢ (𝜑 → 𝑋 ∈ (0[,]1))
cvmlift2lem10.2	⊢ (𝜑 → 𝑌 ∈ (0[,]1))

Assertion

Ref	Expression
cvmlift2lem10	⊢ (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))

Distinct variable groups:   𝑐,𝑑,𝑓,𝑘,𝑠,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧,𝐹   𝜑,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑆,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐽,𝑐,𝑑,𝑓,𝑘,𝑠,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐺,𝑐,𝑓,𝑘,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐻,𝑐,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑋,𝑐,𝑑,𝑓,𝑘,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐶,𝑐,𝑑,𝑓,𝑘,𝑠,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑃,𝑓,𝑘,𝑢,𝑣,𝑥,𝑦,𝑧   𝐵,𝑐,𝑑,𝑣,𝑤,𝑥,𝑦,𝑧   𝑌,𝑐,𝑑,𝑓,𝑘,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐾,𝑐,𝑑,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑘,𝑠,𝑐,𝑑)   𝐵(𝑢,𝑓,𝑘,𝑠)   𝑃(𝑤,𝑠,𝑐,𝑑)   𝑆(𝑘,𝑠,𝑐,𝑑)   𝐺(𝑠,𝑑)   𝐻(𝑘,𝑠,𝑑)   𝐾(𝑘,𝑠)   𝑋(𝑠)   𝑌(𝑠)

Proof of Theorem cvmlift2lem10

Dummy variables 𝑏 𝑚 𝑎 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
2		cvmlift2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))
3		iitop 24829	. . . . . . 7 ⊢ II ∈ Top
4		iiuni 24830	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 3, 4, 4	txunii 23537	. . . . . 6 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
6		eqid 2736	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23190	. . . . 5 ⊢ (𝐺 ∈ ((II ×t II) Cn 𝐽) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽)
8	2, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽)
9		cvmlift2lem10.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
10		cvmlift2lem10.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (0[,]1))
11	9, 10	opelxpd 5663	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
12	8, 11	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ ∪ 𝐽)
13		cvmlift2lem10.s	. . . 4 ⊢ 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))))})
14	13, 6	cvmcov 35457	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ ∪ 𝐽) → ∃𝑚 ∈ 𝐽 ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅))
15	1, 12, 14	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑚 ∈ 𝐽 ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅))
16		n0 4305	. . . . 5 ⊢ ((𝑆‘𝑚) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑚))
17		eleq1 2824	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏)))
18		opelxp 5660	. . . . . . . . . . . . 13 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏) ↔ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏))
19	17, 18	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏)))
20	19	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) ↔ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
21	20	2rexbidv 3201	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (∃𝑎 ∈ II ∃𝑏 ∈ II (𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) ↔ ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
22	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
23	13	cvmsrcl 35458	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑆‘𝑚) → 𝑚 ∈ 𝐽)
24	23	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝑚 ∈ 𝐽)
25		cnima 23209	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ ((II ×t II) Cn 𝐽) ∧ 𝑚 ∈ 𝐽) → (◡𝐺 “ 𝑚) ∈ (II ×t II))
26	22, 24, 25	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (◡𝐺 “ 𝑚) ∈ (II ×t II))
27		eltx 23512	. . . . . . . . . . . 12 ⊢ ((II ∈ Top ∧ II ∈ Top) → ((◡𝐺 “ 𝑚) ∈ (II ×t II) ↔ ∀𝑧 ∈ (◡𝐺 “ 𝑚)∃𝑎 ∈ II ∃𝑏 ∈ II (𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
28	3, 3, 27	mp2an 692	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ 𝑚) ∈ (II ×t II) ↔ ∀𝑧 ∈ (◡𝐺 “ 𝑚)∃𝑎 ∈ II ∃𝑏 ∈ II (𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
29	26, 28	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∀𝑧 ∈ (◡𝐺 “ 𝑚)∃𝑎 ∈ II ∃𝑏 ∈ II (𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
30	11	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
31		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚)
32	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽)
33		ffn 6662	. . . . . . . . . . . 12 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽 → 𝐺 Fn ((0[,]1) × (0[,]1)))
34		elpreima 7003	. . . . . . . . . . . 12 ⊢ (𝐺 Fn ((0[,]1) × (0[,]1)) → (⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚) ↔ (⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)) ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚)))
35	32, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚) ↔ (⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)) ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚)))
36	30, 31, 35	mpbir2and 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚))
37	21, 29, 36	rspcdva 3577	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
38		iillysconn 35447	. . . . . . . . . . . . 13 ⊢ II ∈ Locally SConn
39		simplrl 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → 𝑎 ∈ II)
40		simprll 778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → 𝑋 ∈ 𝑎)
41		llyi 23418	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑎 ∈ II ∧ 𝑋 ∈ 𝑎) → ∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn))
42	38, 39, 40, 41	mp3an2i 1468	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → ∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn))
43		simplrr 777	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → 𝑏 ∈ II)
44		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → 𝑌 ∈ 𝑏)
45		llyi 23418	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑏 ∈ II ∧ 𝑌 ∈ 𝑏) → ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))
46	38, 43, 44, 45	mp3an2i 1468	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))
47		reeanv 3208	. . . . . . . . . . . . 13 ⊢ (∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) ↔ (∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)))
48		simpl2 1193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → 𝑋 ∈ 𝑢)
49	48	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → 𝑋 ∈ 𝑢))
50		simpr2 1196	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → 𝑌 ∈ 𝑣)
51	50	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → 𝑌 ∈ 𝑣))
52		simprl1 1219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) ∧ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))) → 𝑢 ⊆ 𝑎)
53		simprr1 1222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) ∧ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))) → 𝑣 ⊆ 𝑏)
54		xpss12 5639	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 ⊆ 𝑎 ∧ 𝑣 ⊆ 𝑏) → (𝑢 × 𝑣) ⊆ (𝑎 × 𝑏))
55	52, 53, 54	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) ∧ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))) → (𝑢 × 𝑣) ⊆ (𝑎 × 𝑏))
56		simplrr 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) ∧ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))) → (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))
57	55, 56	sstrd 3944	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) ∧ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))) → (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚))
58	57	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)))
59	49, 51, 58	3jcad 1129	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚))))
60		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) → (II ↾t 𝑢) ∈ SConn)
61		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn) → (II ↾t 𝑣) ∈ SConn)
62	60, 61	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))
63	59, 62	jca2 513	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
64	63	reximdv 3151	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (∃𝑣 ∈ II ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
65	64	reximdv 3151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → (∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
66	47, 65	biimtrrid 243	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → ((∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) ∧ ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn)) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
67	42, 46, 66	mp2and 699	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) ∧ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn)))
68	67	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑎 ∈ II ∧ 𝑏 ∈ II)) → (((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
69	68	rexlimdvva 3193	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))))
70	37, 69	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn)))
71		simp3l1 1279	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → 𝑋 ∈ 𝑢)
72		simp3l2 1280	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → 𝑌 ∈ 𝑣)
73		cvmlift2.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ∪ 𝐶
74		simpl1l 1225	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝜑)
75	74, 1	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
76	74, 2	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
77		cvmlift2.p	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
78	74, 77	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑃 ∈ 𝐵)
79		cvmlift2.i	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
80	74, 79	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝐹‘𝑃) = (0𝐺0))
81		cvmlift2.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
82		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))
83		df-ov 7361	. . . . . . . . . . . . . 14 ⊢ (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
84		simpl1r 1226	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚)))
85	84	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚)
86	83, 85	eqeltrid 2840	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝑋𝐺𝑌) ∈ 𝑚)
87	84	simprd 495	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑡 ∈ (𝑆‘𝑚))
88		simpl2l 1227	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑢 ∈ II)
89		simpl2r 1228	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑣 ∈ II)
90		simp3rl 1247	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → (II ↾t 𝑢) ∈ SConn)
91	90	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (II ↾t 𝑢) ∈ SConn)
92		sconnpconn 35421	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ SConn → (II ↾t 𝑢) ∈ PConn)
93		pconnconn 35425	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ PConn → (II ↾t 𝑢) ∈ Conn)
94	91, 92, 93	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (II ↾t 𝑢) ∈ Conn)
95		simp3rr 1248	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → (II ↾t 𝑣) ∈ SConn)
96	95	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (II ↾t 𝑣) ∈ SConn)
97		sconnpconn 35421	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ SConn → (II ↾t 𝑣) ∈ PConn)
98		pconnconn 35425	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ PConn → (II ↾t 𝑣) ∈ Conn)
99	96, 97, 98	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (II ↾t 𝑣) ∈ Conn)
100	71	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑋 ∈ 𝑢)
101	72	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑌 ∈ 𝑣)
102		simp3l3 1281	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚))
103	102	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚))
104		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → 𝑤 ∈ 𝑣)
105		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
106		eqid 2736	. . . . . . . . . . . . 13 ⊢ (℩𝑏 ∈ 𝑡 (𝑋𝐾𝑌) ∈ 𝑏) = (℩𝑏 ∈ 𝑡 (𝑋𝐾𝑌) ∈ 𝑏)
107	73, 75, 76, 78, 80, 81, 82, 13, 86, 87, 88, 89, 94, 99, 100, 101, 103, 104, 105, 106	cvmlift2lem9 35505	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) ∧ (𝑤 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))
108	107	rexlimdvaa 3138	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))
109	71, 72, 108	3jca 1128	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II) ∧ ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn))) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
110	109	3expia 1121	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → (((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn)) → (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
111	110	reximdvva 3184	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (∃𝑢 ∈ II ∃𝑣 ∈ II ((𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐺 “ 𝑚)) ∧ ((II ↾t 𝑢) ∈ SConn ∧ (II ↾t 𝑣) ∈ SConn)) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
112	70, 111	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
113	112	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚) → (𝑡 ∈ (𝑆‘𝑚) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
114	113	exlimdv 1934	. . . . 5 ⊢ ((𝜑 ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚) → (∃𝑡 𝑡 ∈ (𝑆‘𝑚) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
115	16, 114	biimtrid 242	. . . 4 ⊢ ((𝜑 ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚) → ((𝑆‘𝑚) ≠ ∅ → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
116	115	expimpd 453	. . 3 ⊢ (𝜑 → (((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
117	116	rexlimdvw 3142	. 2 ⊢ (𝜑 → (∃𝑚 ∈ 𝐽 ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))))
118	15, 117	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ⟨cop 4586  ∪ cuni 4863   ↦ cmpt 5179   × cxp 5622  ^◡ccnv 5623   ↾ cres 5626   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  ℩crio 7314  (class class class)co 7358   ∈ cmpo 7360  0cc0 11026  1c1 11027  [,]cicc 13264   ↾_t crest 17340  Topctop 22837   Cn ccn 23168  Conncconn 23355  Locally clly 23408   ×_t ctx 23504  Homeochmeo 23697  IIcii 24824  PConncpconn 35413  SConncsconn 35414   CovMap ccvm 35449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-ec 8637  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-sum 15610  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-xrs 17423  df-qtop 17428  df-imas 17429  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-cnfld 21310  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-cn 23171  df-cnp 23172  df-cmp 23331  df-conn 23356  df-lly 23410  df-nlly 23411  df-tx 23506  df-hmeo 23699  df-xms 24264  df-ms 24265  df-tms 24266  df-ii 24826  df-cncf 24827  df-htpy 24925  df-phtpy 24926  df-phtpc 24947  df-pconn 35415  df-sconn 35416  df-cvm 35450

This theorem is referenced by:  cvmlift2lem12  35508