Theorem cnconn 23307

Description: Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Hypothesis

Ref	Expression
cnconn.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
cnconn	⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

Proof of Theorem cnconn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop2 23126	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		df-ne 2926	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
4		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐽 ∈ Conn)
6		simpl3 1194	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹 ∈ (𝐽 Cn 𝐾))
7		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)))
8	7	elin1d 4155	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ 𝐾)
9		cnima 23150	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
10	6, 8, 9	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
11		elssuni 4888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
12	8, 11	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ∪ 𝐾)
13		cnconn.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑌 = ∪ 𝐾
14	12, 13	sseqtrrdi 3977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ 𝑌)
15		simpl2 1193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹:𝑋–onto→𝑌)
16		forn 6739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
17	15, 16	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → ran 𝐹 = 𝑌)
18	14, 17	sseqtrrd 3973	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ran 𝐹)
19		df-rn 5630	. . . . . . . . . . . . . . . 16 ⊢ ran 𝐹 = dom ◡𝐹
20	18, 19	sseqtrdi 3976	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ dom ◡𝐹)
21		sseqin2 4174	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ dom ◡𝐹 ↔ (dom ◡𝐹 ∩ 𝑥) = 𝑥)
22	20, 21	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) = 𝑥)
23		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅)
24	22, 23	eqnetrd 2992	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
25		imadisj 6031	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑥) = ∅ ↔ (dom ◡𝐹 ∩ 𝑥) = ∅)
26	25	necon3bii 2977	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑥) ≠ ∅ ↔ (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
27	24, 26	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ≠ ∅)
28	7	elin2d 4156	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Clsd‘𝐾))
29		cnclima 23153	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
30	6, 28, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
31	4, 5, 10, 27, 30	connclo 23300	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = ∪ 𝐽)
32	4, 13	cnf 23131	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
33		fdm 6661	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶𝑌 → dom 𝐹 = ∪ 𝐽)
34	6, 32, 33	3syl 18	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = ∪ 𝐽)
35		fof 6736	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
36		fdm 6661	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
37	15, 35, 36	3syl 18	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = 𝑋)
38	31, 34, 37	3eqtr2d 2770	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = 𝑋)
39	38	imaeq2d 6011	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ 𝑋))
40		foimacnv 6781	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑥 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
41	15, 14, 40	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
42		foima 6741	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
43	15, 42	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ 𝑋) = 𝑌)
44	39, 41, 43	3eqtr3d 2772	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 = 𝑌)
45	44	expr 456	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 ≠ ∅ → 𝑥 = 𝑌))
46	3, 45	biimtrrid 243	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (¬ 𝑥 = ∅ → 𝑥 = 𝑌))
47	46	orrd 863	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 = ∅ ∨ 𝑥 = 𝑌))
48		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
49	48	elpr 4602	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝑌} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝑌))
50	47, 49	sylibr 234	. . . 4 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → 𝑥 ∈ {∅, 𝑌})
51	50	ex 412	. . 3 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) → 𝑥 ∈ {∅, 𝑌}))
52	51	ssrdv 3941	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌})
53	13	isconn2 23299	. 2 ⊢ (𝐾 ∈ Conn ↔ (𝐾 ∈ Top ∧ (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌}))
54	2, 52, 53	sylanbrc 583	1 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {cpr 4579  ∪ cuni 4858  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  ⟶wf 6478  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349  Topctop 22778  Clsdccld 22901   Cn ccn 23109  Conncconn 23296

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-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  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-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-top 22779  df-topon 22796  df-cld 22904  df-cn 23112  df-conn 23297

This theorem is referenced by:  connima  23310  conncn  23311  qtopconn  23594  connhmph  23674  ivthALT  36319