Theorem cvmlift2lem10 33174

Description: Lemma for cvmlift2 33178. (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 23949	. . . . . . 7 ⊢ II ∈ Top
4		iiuni 23950	. . . . . . 7 ⊢ (0[,]1) = ∪ II
5	3, 3, 4, 4	txunii 22652	. . . . . 6 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
6		eqid 2738	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 22305	. . . . 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 5618	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((0[,]1) × (0[,]1)))
12	8, 11	ffvelrnd 6944	. . 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 33125	. . 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 4277	. . . . 5 ⊢ ((𝑆‘𝑚) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑚))
17		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝑧 ∈ (𝑎 × 𝑏) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑎 × 𝑏)))
18		opelxp 5616	. . . . . . . . . . . . 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 629	. . . . . . . . . . 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 33126	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑆‘𝑚) → 𝑚 ∈ 𝐽)
24	23	ad2antll 725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → 𝑚 ∈ 𝐽)
25		cnima 22324	. . . . . . . . . . . 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 22627	. . . . . . . . . . . 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 688	. . . . . . . . . . 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 767	. . . . . . . . . . 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 6584	. . . . . . . . . . . 12 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝐽 → 𝐺 Fn ((0[,]1) × (0[,]1)))
34		elpreima 6917	. . . . . . . . . . . 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 709	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ⟨𝑋, 𝑌⟩ ∈ (◡𝐺 “ 𝑚))
37	21, 29, 36	rspcdva 3554	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐺‘⟨𝑋, 𝑌⟩) ∈ 𝑚 ∧ 𝑡 ∈ (𝑆‘𝑚))) → ∃𝑎 ∈ II ∃𝑏 ∈ II ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐺 “ 𝑚)))
38		iillysconn 33115	. . . . . . . . . . . . 13 ⊢ II ∈ Locally SConn
39		simplrl 773	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 22533	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑎 ∈ II ∧ 𝑋 ∈ 𝑎) → ∃𝑢 ∈ II (𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn))
42	38, 39, 40, 41	mp3an2i 1464	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 22533	. . . . . . . . . . . . 13 ⊢ ((II ∈ Locally SConn ∧ 𝑏 ∈ II ∧ 𝑌 ∈ 𝑏) → ∃𝑣 ∈ II (𝑣 ⊆ 𝑏 ∧ 𝑌 ∈ 𝑣 ∧ (II ↾t 𝑣) ∈ SConn))
46	38, 43, 44, 45	mp3an2i 1464	. . . . . . . . . . . 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 3292	. . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . . . . 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 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))) → 𝑣 ⊆ 𝑏)
54		xpss12 5595	. . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ⊆ 𝑎 ∧ 𝑋 ∈ 𝑢 ∧ (II ↾t 𝑢) ∈ SConn) → (II ↾t 𝑢) ∈ SConn)
61		simp3 1136	. . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . 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	syl5bir 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 695	. . . . . . . . . . 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 3222	. . . . . . . . 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 1276	. . . . . . . . . . 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 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))) → 𝑌 ∈ 𝑣)
73		cvmlift2.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ∪ 𝐶
74		simpl1l 1222	. . . . . . . . . . . . . 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 7258	. . . . . . . . . . . . . 14 ⊢ (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
84		simpl1r 1223	. . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . 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 1224	. . . . . . . . . . . . 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 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))) → 𝑣 ∈ II)
90		simp3rl 1244	. . . . . . . . . . . . . . 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 33089	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑢) ∈ SConn → (II ↾t 𝑢) ∈ PConn)
93		pconnconn 33093	. . . . . . . . . . . . . 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 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 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 33089	. . . . . . . . . . . . . 14 ⊢ ((II ↾t 𝑣) ∈ SConn → (II ↾t 𝑣) ∈ PConn)
98		pconnconn 33093	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 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))) → (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
106		eqid 2738	. . . . . . . . . . . . 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 33173	. . . . . . . . . . . 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 3213	. . . . . . . . . . 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 1126	. . . . . . . . . 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 1119	. . . . . . . . 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 3205	. . . . . . . 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 1937	. . . . 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	syl5bi 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 3218	. 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 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ⟨cop 4564  ∪ cuni 4836   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  ℩crio 7211  (class class class)co 7255   ∈ cmpo 7257  0cc0 10802  1c1 10803  [,]cicc 13011   ↾_t crest 17048  Topctop 21950   Cn ccn 22283  Conncconn 22470  Locally clly 22523   ×_t ctx 22619  Homeochmeo 22812  IIcii 23944  PConncpconn 33081  SConncsconn 33082   CovMap ccvm 33117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-ec 8458  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-cn 22286  df-cnp 22287  df-cmp 22446  df-conn 22471  df-lly 22525  df-nlly 22526  df-tx 22621  df-hmeo 22814  df-xms 23381  df-ms 23382  df-tms 23383  df-ii 23946  df-htpy 24039  df-phtpy 24040  df-phtpc 24061  df-pconn 33083  df-sconn 33084  df-cvm 33118

This theorem is referenced by:  cvmlift2lem12  33176