Theorem unconn 23437

Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)

Assertion

Ref	Expression
unconn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn))

Proof of Theorem unconn

Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4353	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
2		uniiun 5058	. . . . . . . . 9 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘
3		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 22932	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝑋 ∈ 𝐽)
6		simpl2l 1227	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐴 ⊆ 𝑋)
7	5, 6	ssexd 5324	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8		simpl2r 1228	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐵 ⊆ 𝑋)
9	5, 8	ssexd 5324	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10		uniprg 4923	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
11	7, 9, 10	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
12	2, 11	eqtr3id 2791	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘 = (𝐴 ∪ 𝐵))
13	12	oveq2d 7447	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → (𝐽 ↾t ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘) = (𝐽 ↾t (𝐴 ∪ 𝐵)))
14		vex 3484	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
15	14	elpr 4650	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵))
16		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋))
17		sseq1 4009	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑘 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝐴 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
19		sseq1 4009	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑘 ⊆ 𝑋 ↔ 𝐵 ⊆ 𝑋))
20	19	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝐵 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
21	18, 20	jaoa 958	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝑘 ⊆ 𝑋))
22	16, 21	mpan9 506	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝑘 ⊆ 𝑋)
23	15, 22	sylan2b 594	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑘 ⊆ 𝑋)
24		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
25		elin 3967	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
26	24, 25	sylib 218	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
27		eleq2 2830	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐴))
28	27	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑘))
29		eleq2 2830	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐵))
30	29	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑘))
31	28, 30	jaoa 958	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑘))
32	26, 31	mpan9 506	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝑥 ∈ 𝑘)
33	15, 32	sylan2b 594	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑥 ∈ 𝑘)
34		simpr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn))
35		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐴))
36	35	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐴) ∈ Conn))
37	36	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝐴) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
38		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐵))
39	38	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐵) ∈ Conn))
40	39	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝐵) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
41	37, 40	jaoa 958	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t 𝑘) ∈ Conn))
42	34, 41	mpan9 506	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → (𝐽 ↾t 𝑘) ∈ Conn)
43	15, 42	sylan2b 594	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → (𝐽 ↾t 𝑘) ∈ Conn)
44	3, 23, 33, 43	iunconn 23436	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → (𝐽 ↾t ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘) ∈ Conn)
45	13, 44	eqeltrrd 2842	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn)
46	45	ex 412	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn))
47	46	3expia 1122	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋)) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn)))
48	47	exlimdv 1933	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋)) → (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn)))
49	1, 48	biimtrid 242	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋)) → ((𝐴 ∩ 𝐵) ≠ ∅ → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn)))
50	49	3impia 1118	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {cpr 4628  ∪ cuni 4907  ∪ ciun 4991  ‘cfv 6561  (class class class)co 7431   ↾_t crest 17465  TopOnctopon 22916  Conncconn 23419

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-conn 23420

This theorem is referenced by: (None)