Theorem cvmlift2lem10 34767

Description: Lemma for cvmlift2 34771. (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 24720	. . . . . . 7 ⊢ II ∈ Top
4		iiuni 24721	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 3, 4, 4	txunii 23417	. . . . . 6 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
6		eqid 2731	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23070	. . . . 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 5715	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
12	8, 11	ffvelcdmd 7087	. . 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 34718	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘⟨𝑋, 𝑌⟩) ∈ ∪ 𝐽) → ∃𝑚 ∈ 𝐽 ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅))
15	1, 12, 14	syl2anc 583	. 2 ⊢ (𝜑 → ∃𝑚 ∈ 𝐽 ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ (𝑆‘𝑚) ≠ ∅))
16		n0 4346	. . . . 5 ⊢ ((𝑆‘𝑚) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑚))
17		eleq1 2820	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏)))
18		opelxp 5712	. . . . . . . . . . . . 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 629	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) ↔ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
21	20	2rexbidv 3218	. . . . . . . . . 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 34719	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑆‘𝑚) → 𝑚 ∈ 𝐽)
24	23	ad2antll 726	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝑚 ∈ 𝐽)
25		cnima 23089	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ ((II ×t II) Cn 𝐽) ∧ 𝑚 ∈ 𝐽) → (◡𝐺 “ 𝑚) ∈ (II ×t II))
26	22, 24, 25	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (◡𝐺 “ 𝑚) ∈ (II ×t II))
27		eltx 23392	. . . . . . . . . . . 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 689	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ 𝑚) ∈ (II ×t II) ↔ ∀𝑧 ∈ (◡𝐺 “ 𝑚)∃𝑎 ∈ II ∃𝑏 ∈ II (𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
29	26, 28	sylib 217	. . . . . . . . . 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 768	. . . . . . . . . . 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 6717	. . . . . . . . . . . 12 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽 → 𝐺 Fn ((0[,]1) × (0[,]1)))
34		elpreima 7059	. . . . . . . . . . . 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 710	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚))
37	21, 29, 36	rspcdva 3613	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
38		iillysconn 34708	. . . . . . . . . . . . 13 ⊢ II ∈ Locally SConn
39		simplrl 774	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 23298	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑎 ∈ II ∧ 𝑋 ∈ 𝑎) → ∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn))
42	38, 39, 40, 41	mp3an2i 1465	. . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 23298	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑏 ∈ II ∧ 𝑌 ∈ 𝑏) → ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))
46	38, 43, 44, 45	mp3an2i 1465	. . . . . . . . . . . 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 3225	. . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 ⊆ 𝑎 ∧ 𝑣 ⊆ 𝑏) → (𝑢 × 𝑣) ⊆ (𝑎 × 𝑏))
55	52, 53, 54	syl2anc 583	. . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) → (II ↾t 𝑢) ∈ SConn)
61		simp3 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn) → (II ↾t 𝑣) ∈ SConn)
62	60, 61	anim12i 612	. . . . . . . . . . . . . . . 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 3169	. . . . . . . . . . . . . 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 3169	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 696	. . . . . . . . . . 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 3210	. . . . . . . . 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 1277	. . . . . . . . . . 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 1278	. . . . . . . . . . 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 1223	. . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . 14 ⊢ (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
84		simpl1r 1224	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 1225	. . . . . . . . . . . . 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 1226	. . . . . . . . . . . . 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 1245	. . . . . . . . . . . . . . 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 34682	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ SConn → (II ↾t 𝑢) ∈ PConn)
93		pconnconn 34686	. . . . . . . . . . . . . 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 1246	. . . . . . . . . . . . . . 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 34682	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ SConn → (II ↾t 𝑣) ∈ PConn)
98		pconnconn 34686	. . . . . . . . . . . . . 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 1279	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 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))) → (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
106		eqid 2731	. . . . . . . . . . . . 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 34766	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 1127	. . . . . . . . . 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 1120	. . . . . . . . 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 3204	. . . . . . . 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 1935	. . . . 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 241	. . . 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 3159	. 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 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3431   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ⟨cop 4634  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675   ↾ cres 5678   “ cima 5679   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  ℩crio 7367  (class class class)co 7412   ∈ cmpo 7414  0cc0 11116  1c1 11117  [,]cicc 13334   ↾_t crest 17373  Topctop 22715   Cn ccn 23048  Conncconn 23235  Locally clly 23288   ×_t ctx 23384  Homeochmeo 23577  IIcii 24715  PConncpconn 34674  SConncsconn 34675   CovMap ccvm 34710

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195  ax-mulf 11196

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-ec 8711  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-fi 9412  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ioo 13335  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-rest 17375  df-topn 17376  df-0g 17394  df-gsum 17395  df-topgen 17396  df-pt 17397  df-prds 17400  df-xrs 17455  df-qtop 17460  df-imas 17461  df-xps 17463  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-mulg 18994  df-cntz 19229  df-cmn 19698  df-psmet 21225  df-xmet 21226  df-met 21227  df-bl 21228  df-mopn 21229  df-cnfld 21234  df-top 22716  df-topon 22733  df-topsp 22755  df-bases 22769  df-cld 22843  df-ntr 22844  df-cls 22845  df-nei 22922  df-cn 23051  df-cnp 23052  df-cmp 23211  df-conn 23236  df-lly 23290  df-nlly 23291  df-tx 23386  df-hmeo 23579  df-xms 24146  df-ms 24147  df-tms 24148  df-ii 24717  df-htpy 24816  df-phtpy 24817  df-phtpc 24838  df-pconn 34676  df-sconn 34677  df-cvm 34711

This theorem is referenced by:  cvmlift2lem12  34769