Theorem cvmlift2lem10 33906

Description: Lemma for cvmlift2 33910. (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 24243	. . . . . . 7 ⊢ II ∈ Top
4		iiuni 24244	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 3, 4, 4	txunii 22944	. . . . . 6 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
6		eqid 2736	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 22597	. . . . 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 5671	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
12	8, 11	ffvelcdmd 7036	. . 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 33857	. . 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 4306	. . . . 5 ⊢ ((𝑆‘𝑚) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑚))
17		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏)))
18		opelxp 5669	. . . . . . . . . . . . 13 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏) ↔ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏))
19	17, 18	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏)))
20	19	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) ↔ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
21	20	2rexbidv 3213	. . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
23	13	cvmsrcl 33858	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑆‘𝑚) → 𝑚 ∈ 𝐽)
24	23	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝑚 ∈ 𝐽)
25		cnima 22616	. . . . . . . . . . . 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 22919	. . . . . . . . . . . 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 690	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
31		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → (𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚)
32	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽)
33		ffn 6668	. . . . . . . . . . . 12 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽 → 𝐺 Fn ((0[,]1) × (0[,]1)))
34		elpreima 7008	. . . . . . . . . . . 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 711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚))
37	21, 29, 36	rspcdva 3582	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
38		iillysconn 33847	. . . . . . . . . . . . 13 ⊢ II ∈ Locally SConn
39		simplrl 775	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 22825	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑎 ∈ II ∧ 𝑋 ∈ 𝑎) → ∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn))
42	38, 39, 40, 41	mp3an2i 1466	. . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . 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 22825	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑏 ∈ II ∧ 𝑌 ∈ 𝑏) → ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))
46	38, 43, 44, 45	mp3an2i 1466	. . . . . . . . . . . 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 3217	. . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . 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 1195	. . . . . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . 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 697	. . . . . . . . . . 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 413	. . . . . . . . . 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 3205	. . . . . . . . 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 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))) → 𝑋 ∈ 𝑢)
72		simp3l2 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))) → 𝑌 ∈ 𝑣)
73		cvmlift2.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ∪ 𝐶
74		simpl1l 1224	. . . . . . . . . . . . . 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 7360	. . . . . . . . . . . . . 14 ⊢ (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
84		simpl1r 1225	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 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))) → 𝑢 ∈ II)
89		simpl2r 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))) → 𝑣 ∈ II)
90		simp3rl 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 𝑢) ∈ SConn)
91	90	adantr 481	. . . . . . . . . . . . . 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 33821	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ SConn → (II ↾t 𝑢) ∈ PConn)
93		pconnconn 33825	. . . . . . . . . . . . . 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 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 𝑣) ∈ SConn)
96	95	adantr 481	. . . . . . . . . . . . . 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 33821	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ SConn → (II ↾t 𝑣) ∈ PConn)
98		pconnconn 33825	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 1280	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 33905	. . . . . . . . . . . 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 3153	. . . . . . . . . . 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 3202	. . . . . . . 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 457	. . . . . 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 1936	. . . . 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 454	. . 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 3157	. 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 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ⟨cop 4592  ∪ cuni 4865   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635   “ cima 5636   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  ℩crio 7312  (class class class)co 7357   ∈ cmpo 7359  0cc0 11051  1c1 11052  [,]cicc 13267   ↾_t crest 17302  Topctop 22242   Cn ccn 22575  Conncconn 22762  Locally clly 22815   ×_t ctx 22911  Homeochmeo 23104  IIcii 24238  PConncpconn 33813  SConncsconn 33814   CovMap ccvm 33849

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-ec 8650  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-cn 22578  df-cnp 22579  df-cmp 22738  df-conn 22763  df-lly 22817  df-nlly 22818  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-ii 24240  df-htpy 24333  df-phtpy 24334  df-phtpc 24355  df-pconn 33815  df-sconn 33816  df-cvm 33850

This theorem is referenced by:  cvmlift2lem12  33908