connpconn - Mathbox for Mario Carneiro

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Theorem connpconn 35212

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 23302	. . 3 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ Top)
3		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		simpll 766	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Conn)
5		inss1 4188	. . . . . . 7 ⊢ (𝐽 ∩ (Clsd‘𝐽)) ⊆ 𝐽
6		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝐽 ∈ 𝑛-Locally PConn)
7	1	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝐽 ∈ Top)
8	3	topopn 22791	. . . . . . . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑧 ∈ ∪ 𝐽)) → 𝑧 ∈ ∪ 𝐽)
11		nlly2i 23361	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 1222	. . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑤))
15	14	anbi2d 630	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑤 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤)))
16	15	rexbidv 3153	. . . . . . . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 ∈ 𝒫 ∪ 𝐽 → 𝑠 ⊆ ∪ 𝐽)
27	26	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 23047	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ∪ (𝐽 ↾_t 𝑠) = ∪ (𝐽 ↾_t 𝑠)
36	35	pconncn 35201	. . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢) → 𝑔 ∈ (II Cn 𝐽))
39	38	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23172	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑤 ∈ 𝑢 ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))) ∧ 𝑦 ∈ 𝑢) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))
46	45	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . . 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 24915	. . . . . . . . . . . . . . . . . . . . . . . . 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 24912	. . . . . . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . 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 24913	. . . . . . . . . . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → (𝑓‘0) = ((𝑔(_𝑝‘𝐽)ℎ)‘0))
58	57	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → ((𝑓‘0) = 𝑥 ↔ ((𝑔(_𝑝‘𝐽)ℎ)‘0) = 𝑥))
59		fveq1 6821	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → (𝑓‘1) = ((𝑔(_𝑝‘𝐽)ℎ)‘1))
60	59	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑔(_𝑝‘𝐽)ℎ) → ((𝑓‘1) = 𝑦 ↔ ((𝑔(_𝑝‘𝐽)ℎ)‘1) = 𝑦))
61	58, 60	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . . . . . 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 3136	. . . . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓‘0) = (𝑔‘0))
75	74	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓‘0) = 𝑥 ↔ (𝑔‘0) = 𝑥))
76		fveq1 6821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓‘1) = (𝑔‘1))
77	76	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓‘1) = 𝑤 ↔ (𝑔‘1) = 𝑤))
78	75, 77	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤) ↔ ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤)))
79	78	cbvrexvw 3208	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑤) ↔ ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑤))
80		ssrab 4024	. . . . . . . . . . . . . . . . . 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 296	. . . . . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 3144	. . . . . . . . . . . 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 3130	. . . . . . . . . . 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 467	. . . . . . . . 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 3121	. . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top)
92		ssrab2 4031	. . . . . . . . 9 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ⊆ ∪ 𝐽
93	3	isclo2 22973	. . . . . . . . 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 257	. . . . . . 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 3933	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} ∈ 𝐽)
97		simpr 484	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
98		iitopon 24770	. . . . . . . . . 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 22802	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
101	91, 100	sylib 218	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ (TopOn‘∪ 𝐽))
102		cnconst2 23168	. . . . . . . . 9 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ((0[,]1) × {𝑥}) ∈ (II Cn 𝐽))
103	99, 101, 97, 102	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ((0[,]1) × {𝑥}) ∈ (II Cn 𝐽))
104		0elunit 13372	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
105		vex 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
106	105	fvconst2 7140	. . . . . . . . 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 13373	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
109	105	fvconst2 7140	. . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑥))
112	111	anbi2d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑥)))
113		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → (𝑓‘0) = (((0[,]1) × {𝑥})‘0))
114	113	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → ((𝑓‘0) = 𝑥 ↔ (((0[,]1) × {𝑥})‘0) = 𝑥))
115		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → (𝑓‘1) = (((0[,]1) × {𝑥})‘1))
116	115	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = ((0[,]1) × {𝑥}) → ((𝑓‘1) = 𝑥 ↔ (((0[,]1) × {𝑥})‘1) = 𝑥))
117	114, 116	anbi12d 632	. . . . . . . . 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 3590	. . . . . . . 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 1376	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
120		rabn0 4340	. . . . . . 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 234	. . . . . 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 3933	. . . . . 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 23300	. . . . 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 2735	. . . 4 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 = {𝑦 ∈ ∪ 𝐽 ∣ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)})
126		rabid2 3428	. . . 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 218	. . 3 ⊢ (((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) ∧ 𝑥 ∈ ∪ 𝐽) → ∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
128	127	ralrimiva 3121	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
129	3	ispconn 35200	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3394 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 𝒫 cpw 4551 {csn 4577 ∪ cuni 4858 × cxp 5617 ‘cfv 6482 (class class class)co 7349 0cc0 11009 1c1 11010 [,]cicc 13251 ↾_t crest 17324 Topctop 22778 TopOnctopon 22795 Clsdccld 22901 Cn ccn 23109 Conncconn 23296 𝑛-Locally cnlly 23350 IIcii 24766 *_𝑝cpco 24898 PConncpconn 35196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-nei 22983 df-cn 23112 df-cnp 23113 df-conn 23297 df-nlly 23352 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-ii 24768 df-pco 24903 df-pconn 35198

This theorem is referenced by: (None)

W3C validator