connpconn - Mathbox for Mario Carneiro

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Theorem connpconn 33829

Description: A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)

**Assertion**
Ref	Expression
connpconn	⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn)

Proof of Theorem connpconn Dummy variables 𝑥 𝑓 𝑦 𝑧 𝑔 ℎ 𝑠 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		conntop 22768	. . 3 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
2	1	adantr 481	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ Top)
3		eqid 2736	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		simpll 765	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Conn)
5		inss1 4188	. . . . . . 7 ⊢ (𝐽 ∩ (Clsd‘𝐽)) ⊆ 𝐽
6		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝐽 ∈ 𝑛-Locally PConn)
7	1	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝐽 ∈ Top)
8	3	topopn 22255	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
9	7, 8	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → ∪ 𝐽 ∈ 𝐽)
10		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝑧 ∈ ∪ 𝐽)
11		nlly2i 22827	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ 𝑛-Locally PConn ∧ ∪ 𝐽 ∈ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽) → ∃𝑠 ∈ 𝒫 ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))
12	6, 9, 10, 11	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → ∃𝑠 ∈ 𝒫 ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))
13		simprr1 1221	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → 𝑧 ∈ 𝑢)
14		eqeq2 2748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑤))
15	14	anbi2d 629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑤 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤)))
16	15	rexbidv 3175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤)))
17	16	elrab 3645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ↔ (𝑤 ∈ ∪ 𝐽 ∧ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤)))
18	17	simprbi 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤))
19		simprr3 1223	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → (𝐽 ↾_t 𝑠) ∈ PConn)
20	19	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → (𝐽 ↾_t 𝑠) ∈ PConn)
21		simprr2 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → 𝑢 ⊆ 𝑠)
22	21	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑢 ⊆ 𝑠)
23		simprll 777	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑤 ∈ 𝑢)
24	22, 23	sseldd 3945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑤 ∈ 𝑠)
25	7	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝐽 ∈ Top)
26		elpwi 4567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 ∈ 𝒫 ∪ 𝐽 → 𝑠 ⊆ ∪ 𝐽)
27	26	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → 𝑠 ⊆ ∪ 𝐽)
28	27	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑠 ⊆ ∪ 𝐽)
29	3	restuni 22513	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → 𝑠 = ∪ (𝐽 ↾_t 𝑠))
30	25, 28, 29	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑠 = ∪ (𝐽 ↾_t 𝑠))
31	24, 30	eleqtrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑤 ∈ ∪ (𝐽 ↾_t 𝑠))
32		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑦 ∈ 𝑢)
33	22, 32	sseldd 3945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑦 ∈ 𝑠)
34	33, 30	eleqtrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → 𝑦 ∈ ∪ (𝐽 ↾_t 𝑠))
35		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ∪ (𝐽 ↾_t 𝑠) = ∪ (𝐽 ↾_t 𝑠)
36	35	pconncn 33818	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐽 ↾_t 𝑠) ∈ PConn ∧ 𝑤 ∈ ∪ (𝐽 ↾_t 𝑠) ∧ 𝑦 ∈ ∪ (𝐽 ↾_t 𝑠)) → ∃ℎ ∈ (II Cn (𝐽 ↾_t 𝑠))((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))
37	20, 31, 34, 36	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → ∃ℎ ∈ (II Cn (𝐽 ↾_t 𝑠))((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))
38		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢) → 𝑔 ∈ (II Cn 𝐽))
39	38	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → 𝑔 ∈ (II Cn 𝐽))
40	25	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → 𝐽 ∈ Top)
41		cnrest2r 22638	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐽 ∈ Top → (II Cn (𝐽 ↾_t 𝑠)) ⊆ (II Cn 𝐽))
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (II Cn (𝐽 ↾_t 𝑠)) ⊆ (II Cn 𝐽))
43		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)))
44	42, 43	sseldd 3945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ℎ ∈ (II Cn 𝐽))
45		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))
46	45	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))
47	46	simprd 496	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (𝑔‘1) = 𝑤)
48		simprrl 779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (ℎ‘0) = 𝑤)
49	47, 48	eqtr4d 2779	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (𝑔‘1) = (ℎ‘0))
50	39, 44, 49	pcocn 24380	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (𝑔(*_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
51	39, 44	pco0 24377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ((𝑔(*_𝑝‘𝐽)ℎ)‘0) = (𝑔‘0))
52	46	simpld 495	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (𝑔‘0) = 𝑥)
53	51, 52	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ((𝑔(*_𝑝‘𝐽)ℎ)‘0) = 𝑥)
54	39, 44	pco1 24378	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ((𝑔(*_𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
55		simprrr 780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → (ℎ‘1) = 𝑦)
56	54, 55	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ((𝑔(*_𝑝‘𝐽)ℎ)‘1) = 𝑦)
57		fveq1 6841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → (𝑓‘0) = ((𝑔(_𝑝‘𝐽)ℎ)‘0))
58	57	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → ((𝑓‘0) = 𝑥 ↔ ((𝑔(_𝑝‘𝐽)ℎ)‘0) = 𝑥))
59		fveq1 6841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → (𝑓‘1) = ((𝑔(_𝑝‘𝐽)ℎ)‘1))
60	59	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → ((𝑓‘1) = 𝑦 ↔ ((𝑔(_𝑝‘𝐽)ℎ)‘1) = 𝑦))
61	58, 60	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑔(_𝑝‘𝐽)ℎ)‘0) = 𝑥 ∧ ((𝑔(*_𝑝‘𝐽)ℎ)‘1) = 𝑦)))
62	61	rspcev 3581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑔(_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ (((𝑔(_𝑝‘𝐽)ℎ)‘0) = 𝑥 ∧ ((𝑔(*_𝑝‘𝐽)ℎ)‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
63	50, 53, 56, 62	syl12anc 835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) ∧ (ℎ ∈ (II Cn (𝐽 ↾_t 𝑠)) ∧ ((ℎ‘0) = 𝑤 ∧ (ℎ‘1) = 𝑦))) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
64	37, 63	rexlimddv 3158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ ((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢)) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
65	64	anassrs 468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ (𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤)))) ∧ 𝑦 ∈ 𝑢) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
66	65	ralrimiva 3143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ (𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤)))) → ∀𝑦 ∈ 𝑢 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
67	66	anassrs 468	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) → ∀𝑦 ∈ 𝑢 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
68	67	rexlimdvaa 3153	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → (∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤) → ∀𝑦 ∈ 𝑢 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
69	21	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → 𝑢 ⊆ 𝑠)
70		simplrl 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → 𝑠 ∈ 𝒫 ∪ 𝐽)
71	70, 26	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → 𝑠 ⊆ ∪ 𝐽)
72	69, 71	sstrd 3954	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → 𝑢 ⊆ ∪ 𝐽)
73	68, 72	jctild 526	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → (∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤) → (𝑢 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝑢 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))))
74		fveq1 6841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0))
75	74	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓‘0) = 𝑥 ↔ (𝑔‘0) = 𝑥))
76		fveq1 6841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓‘1) = (𝑔‘1))
77	76	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓‘1) = 𝑤 ↔ (𝑔‘1) = 𝑤))
78	75, 77	anbi12d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤) ↔ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤)))
79	78	cbvrexvw 3226	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤) ↔ ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))
80		ssrab 4030	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ↔ (𝑢 ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝑢 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
81	73, 79, 80	3imtr4g 295	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤) → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))
82	18, 81	syl5 34	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) ∧ 𝑤 ∈ 𝑢) → (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))
83	82	ralrimiva 3143	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))
84	13, 83	jca 512	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ (𝑠 ∈ 𝒫 ∪ 𝐽 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn))) → (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})))
85	84	expr 457	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ 𝑠 ∈ 𝒫 ∪ 𝐽) → ((𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn) → (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))))
86	85	reximdv 3167	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) ∧ 𝑠 ∈ 𝒫 ∪ 𝐽) → (∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn) → ∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))))
87	86	rexlimdva 3152	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → (∃𝑠 ∈ 𝒫 ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾_t 𝑠) ∈ PConn) → ∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))))
88	12, 87	mpd 15	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → ∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})))
89	88	anassrs 468	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) ∧ 𝑧 ∈ ∪ 𝐽) → ∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})))
90	89	ralrimiva 3143	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∀𝑧 ∈ ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})))
91	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top)
92		ssrab2 4037	. . . . . . . . 9 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ⊆ ∪ 𝐽
93	3	isclo2 22439	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ⊆ ∪ 𝐽) → ({𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑧 ∈ ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))))
94	91, 92, 93	sylancl 586	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ({𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑧 ∈ ∪ 𝐽∃𝑢 ∈ 𝐽 (𝑧 ∈ 𝑢 ∧ ∀𝑤 ∈ 𝑢 (𝑤 ∈ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} → 𝑢 ⊆ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}))))
95	90, 94	mpbird 256	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ (𝐽 ∩ (Clsd‘𝐽)))
96	5, 95	sselid 3942	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ 𝐽)
97		simpr 485	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
98		iitopon 24242	. . . . . . . . . 10 ⊢ II ∈ (TopOn‘(0[,]1))
99	98	a1i 11	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → II ∈ (TopOn‘(0[,]1)))
100	3	toptopon 22266	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
101	91, 100	sylib 217	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐽))
102		cnconst2 22634	. . . . . . . . 9 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ((0[,]1) × {𝑥}) ∈ (II Cn 𝐽))
103	99, 101, 97, 102	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ((0[,]1) × {𝑥}) ∈ (II Cn 𝐽))
104		0elunit 13386	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
105		vex 3449	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
106	105	fvconst2 7153	. . . . . . . . 9 ⊢ (0 ∈ (0[,]1) → (((0[,]1) × {𝑥})‘0) = 𝑥)
107	104, 106	mp1i 13	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → (((0[,]1) × {𝑥})‘0) = 𝑥)
108		1elunit 13387	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
109	105	fvconst2 7153	. . . . . . . . 9 ⊢ (1 ∈ (0[,]1) → (((0[,]1) × {𝑥})‘1) = 𝑥)
110	108, 109	mp1i 13	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → (((0[,]1) × {𝑥})‘1) = 𝑥)
111		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑥))
112	111	anbi2d 629	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑥)))
113		fveq1 6841	. . . . . . . . . . 11 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → (𝑓‘0) = (((0[,]1) × {𝑥})‘0))
114	113	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → ((𝑓‘0) = 𝑥 ↔ (((0[,]1) × {𝑥})‘0) = 𝑥))
115		fveq1 6841	. . . . . . . . . . 11 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → (𝑓‘1) = (((0[,]1) × {𝑥})‘1))
116	115	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → ((𝑓‘1) = 𝑥 ↔ (((0[,]1) × {𝑥})‘1) = 𝑥))
117	114, 116	anbi12d 631	. . . . . . . . 9 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑥) ↔ ((((0[,]1) × {𝑥})‘0) = 𝑥 ∧ (((0[,]1) × {𝑥})‘1) = 𝑥)))
118	112, 117	rspc2ev 3592	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ 𝐽 ∧ ((0[,]1) × {𝑥}) ∈ (II Cn 𝐽) ∧ ((((0[,]1) × {𝑥})‘0) = 𝑥 ∧ (((0[,]1) × {𝑥})‘1) = 𝑥)) → ∃𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119	97, 103, 107, 110, 118	syl112anc 1374	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
120		rabn0 4345	. . . . . . 7 ⊢ ({𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
121	119, 120	sylibr 233	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ≠ ∅)
122		inss2 4189	. . . . . . 7 ⊢ (𝐽 ∩ (Clsd‘𝐽)) ⊆ (Clsd‘𝐽)
123	122, 95	sselid 3942	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ (Clsd‘𝐽))
124	3, 4, 96, 121, 123	connclo 22766	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} = ∪ 𝐽)
125	124	eqcomd 2742	. . . 4 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 = {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})
126		rabid2 3436	. . . 4 ⊢ (∪ 𝐽 = {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ↔ ∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
127	125, 126	sylib 217	. . 3 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
128	127	ralrimiva 3143	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
129	3	ispconn 33817	. 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
130	2, 128, 129	sylanbrc 583	1 ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {csn 4586 ∪ cuni 4865 × cxp 5631 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 [,]cicc 13267 ↾_t crest 17302 Topctop 22242 TopOnctopon 22259 Clsdccld 22367 Cn ccn 22575 Conncconn 22762 𝑛-Locally cnlly 22816 IIcii 24238 *_𝑝cpco 24363 PConncpconn 33813

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-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-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-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-icc 13271 df-fz 13425 df-fzo 13568 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-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-nei 22449 df-cn 22578 df-cnp 22579 df-conn 22763 df-nlly 22818 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-ii 24240 df-pco 24368 df-pconn 33815

This theorem is referenced by: (None)

W3C validator