Theorem unconn 23412

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 4281	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
2		uniiun 4988	. . . . . . . . 9 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑘 ∈ {𝐴, 𝐵}𝑘
3		simpl1 1198	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 22909	. . . . . . . . . . . 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 1233	. . . . . . . . . . 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 5252	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8		simpl2r 1234	. . . . . . . . . . 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 5252	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10		uniprg 4854	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
11	7, 9, 10	syl2anc 590	. . . . . . . . 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 2788	. . . . . . . 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 7372	. . . . . . 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 3435	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
15	14	elpr 4580	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵))
16		simpl2 1199	. . . . . . . . . 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 3940	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑘 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝐴 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
19		sseq1 3940	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑘 ⊆ 𝑋 ↔ 𝐵 ⊆ 𝑋))
20	19	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝐵 ⊆ 𝑋 → 𝑘 ⊆ 𝑋))
21	18, 20	jaoa 963	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝑘 ⊆ 𝑋))
22	16, 21	mpan9 511	. . . . . . . . 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 600	. . . . . . . 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 1200	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
25		elin 3899	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
26	24, 25	sylib 219	. . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐴))
28	27	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑘))
29		eleq2 2828	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝑘 ↔ 𝑥 ∈ 𝐵))
30	29	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑘))
31	28, 30	jaoa 963	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑘))
32	26, 31	mpan9 511	. . . . . . . . 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 600	. . . . . . . 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 7364	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐴))
36	35	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐴) ∈ Conn))
37	36	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ((𝐽 ↾t 𝐴) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
38		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝐵))
39	38	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝑘) ∈ Conn ↔ (𝐽 ↾t 𝐵) ∈ Conn))
40	39	biimprd 249	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ((𝐽 ↾t 𝐵) ∈ Conn → (𝐽 ↾t 𝑘) ∈ Conn))
41	37, 40	jaoa 963	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐴 ∨ 𝑘 = 𝐵) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t 𝑘) ∈ Conn))
42	34, 41	mpan9 511	. . . . . . . . 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 600	. . . . . . . 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 23411	. . . . . . 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 2840	. . . . . 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 1127	. . . 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 1940	. . 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 243	. 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 1123	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 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  {cpr 4557  ∪ cuni 4838  ∪ ciun 4921  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  TopOnctopon 22893  Conncconn 23394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17376  df-topgen 17397  df-top 22877  df-topon 22894  df-bases 22929  df-cld 23002  df-conn 23395

This theorem is referenced by: (None)