Theorem unconn 23339

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 4298	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
2		uniiun 5002	. . . . . . . . 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 22836	. . . . . . . . . . . 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 5257	. . . . . . . . . 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 5257	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10		uniprg 4870	. . . . . . . . . 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 2780	. . . . . . . 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 7357	. . . . . . 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 3440	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
15	14	elpr 4596	. . . . . . . . 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 3955	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑘 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝐴 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
19		sseq1 3955	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑘 ⊆ 𝑋 ↔ 𝐵 ⊆ 𝑋))
20	19	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝐵 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
21	18, 20	jaoa 957	. . . . . . . . . 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 3913	. . . . . . . . . . 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 2820	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐴))
28	27	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑘))
29		eleq2 2820	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐵))
30	29	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑘))
31	28, 30	jaoa 957	. . . . . . . . . 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 7349	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐴))
36	35	eleq1d 2816	. . . . . . . . . . . 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 7349	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐵))
39	38	eleq1d 2816	. . . . . . . . . . . 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 957	. . . . . . . . . 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 23338	. . . . . . 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 2832	. . . . . 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 1121	. . . 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 1934	. . 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 1117	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 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4278  {cpr 4573  ∪ cuni 4854  ∪ ciun 4936  ‘cfv 6476  (class class class)co 7341   ↾_t crest 17319  TopOnctopon 22820  Conncconn 23321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-en 8865  df-fin 8868  df-fi 9290  df-rest 17321  df-topgen 17342  df-top 22804  df-topon 22821  df-bases 22856  df-cld 22929  df-conn 23322

This theorem is referenced by: (None)