Theorem cvmlift2lem10 35280

Description: Lemma for cvmlift2 35284. (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 24925	. . . . . . 7 ⊢ II ∈ Top
4		iiuni 24926	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 3, 4, 4	txunii 23622	. . . . . 6 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
6		eqid 2740	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23275	. . . . 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 5739	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
12	8, 11	ffvelcdmd 7119	. . 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 35231	. . 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 4376	. . . . 5 ⊢ ((𝑆‘𝑚) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑚))
17		eleq1 2832	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏)))
18		opelxp 5736	. . . . . . . . . . . . 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 630	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝑧 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)) ↔ ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚))))
21	20	2rexbidv 3228	. . . . . . . . . 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 35232	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑆‘𝑚) → 𝑚 ∈ 𝐽)
24	23	ad2antll 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝑚 ∈ 𝐽)
25		cnima 23294	. . . . . . . . . . . 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 23597	. . . . . . . . . . . 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 691	. . . . . . . . . . 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 6747	. . . . . . . . . . . 12 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽 → 𝐺 Fn ((0[,]1) × (0[,]1)))
34		elpreima 7091	. . . . . . . . . . . 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 712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚))
37	21, 29, 36	rspcdva 3636	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
38		iillysconn 35221	. . . . . . . . . . . . 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 23503	. . . . . . . . . . . . 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 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 23503	. . . . . . . . . . . . 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 3235	. . . . . . . . . . . . 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 5715	. . . . . . . . . . . . . . . . . . . 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 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 4019	. . . . . . . . . . . . . . . . . 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 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 3176	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . 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 698	. . . . . . . . . . 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 3219	. . . . . . . . 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 7451	. . . . . . . . . . . . . 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 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 2848	. . . . . . . . . . . . 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 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 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 35195	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ SConn → (II ↾t 𝑢) ∈ PConn)
93		pconnconn 35199	. . . . . . . . . . . . . 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 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 35195	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ SConn → (II ↾t 𝑣) ∈ PConn)
98		pconnconn 35199	. . . . . . . . . . . . . 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 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 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 2740	. . . . . . . . . . . . 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 35279	. . . . . . . . . . . 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 3162	. . . . . . . . . . 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 3213	. . . . . . . 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 1932	. . . . 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 3166	. 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 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ⟨cop 4654  ∪ cuni 4931   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699   ↾ cres 5702   “ cima 5703   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  ℩crio 7403  (class class class)co 7448   ∈ cmpo 7450  0cc0 11184  1c1 11185  [,]cicc 13410   ↾_t crest 17480  Topctop 22920   Cn ccn 23253  Conncconn 23440  Locally clly 23493   ×_t ctx 23589  Homeochmeo 23782  IIcii 24920  PConncpconn 35187  SConncsconn 35188   CovMap ccvm 35223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-ec 8765  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-0g 17501  df-gsum 17502  df-topgen 17503  df-pt 17504  df-prds 17507  df-xrs 17562  df-qtop 17567  df-imas 17568  df-xps 17570  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-cntz 19357  df-cmn 19824  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-cnfld 21388  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-cn 23256  df-cnp 23257  df-cmp 23416  df-conn 23441  df-lly 23495  df-nlly 23496  df-tx 23591  df-hmeo 23784  df-xms 24351  df-ms 24352  df-tms 24353  df-ii 24922  df-cncf 24923  df-htpy 25021  df-phtpy 25022  df-phtpc 25043  df-pconn 35189  df-sconn 35190  df-cvm 35224

This theorem is referenced by:  cvmlift2lem12  35282