Theorem unconn 22780

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 4306	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
2		uniiun 5018	. . . . . . . . 9 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘
3		simpl1 1191	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 22275	. . . . . . . . . . . 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 1226	. . . . . . . . . . 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 5281	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8		simpl2r 1227	. . . . . . . . . . 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 5281	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10		uniprg 4882	. . . . . . . . . 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 2790	. . . . . . . 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 7373	. . . . . . 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 3449	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
15	14	elpr 4609	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵))
16		simpl2 1192	. . . . . . . . . 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 3969	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑘 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝐴 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
19		sseq1 3969	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑘 ⊆ 𝑋 ↔ 𝐵 ⊆ 𝑋))
20	19	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝐵 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
21	18, 20	jaoa 954	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝑘 ⊆ 𝑋))
22	16, 21	mpan9 507	. . . . . . . . 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 1193	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
25		elin 3926	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
26	24, 25	sylib 217	. . . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐴))
28	27	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑘))
29		eleq2 2826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐵))
30	29	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑘))
31	28, 30	jaoa 954	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑘))
32	26, 31	mpan9 507	. . . . . . . . 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 485	. . . . . . . . . 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 7365	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐴))
36	35	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐴) ∈ Conn))
37	36	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝐴) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
38		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐵))
39	38	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐵) ∈ Conn))
40	39	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝐵) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
41	37, 40	jaoa 954	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t 𝑘) ∈ Conn))
42	34, 41	mpan9 507	. . . . . . . . 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 22779	. . . . . . 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 2839	. . . . . 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 413	. . . . 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 1936	. . 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 241	. 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {cpr 4588  ∪ cuni 4865  ∪ ciun 4954  ‘cfv 6496  (class class class)co 7357   ↾_t crest 17302  TopOnctopon 22259  Conncconn 22762

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

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-ral 3065  df-rex 3074  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-op 4593  df-uni 4866  df-int 4908  df-iun 4956  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-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-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-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-en 8884  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-conn 22763

This theorem is referenced by: (None)