Theorem cnconn 23451

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 23270	. . 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 2947	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
4		eqid 2740	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5		simpl1 1191	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐽 ∈ Conn)
6		simpl3 1193	. . . . . . . . . . . . 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 4227	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ 𝐾)
9		cnima 23294	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
10	6, 8, 9	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
11		elssuni 4961	. . . . . . . . . . . . . . . . . . 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 4060	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ 𝑌)
15		simpl2 1192	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹:𝑋–onto→𝑌)
16		forn 6837	. . . . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ran 𝐹)
19		df-rn 5711	. . . . . . . . . . . . . . . 16 ⊢ ran 𝐹 = dom ◡𝐹
20	18, 19	sseqtrdi 4059	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ dom ◡𝐹)
21		sseqin2 4244	. . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
25		imadisj 6109	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑥) = ∅ ↔ (dom ◡𝐹 ∩ 𝑥) = ∅)
26	25	necon3bii 2999	. . . . . . . . . . . . 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 4228	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Clsd‘𝐾))
29		cnclima 23297	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
30	6, 28, 29	syl2anc 583	. . . . . . . . . . . 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 23444	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = ∪ 𝐽)
32	4, 13	cnf 23275	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
33		fdm 6756	. . . . . . . . . . . 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 6834	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
36		fdm 6756	. . . . . . . . . . . 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 2786	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = 𝑋)
39	38	imaeq2d 6089	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ 𝑋))
40		foimacnv 6879	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑥 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
41	15, 14, 40	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
42		foima 6839	. . . . . . . . . 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 2788	. . . . . . . 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 862	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 = ∅ ∨ 𝑥 = 𝑌))
48		vex 3492	. . . . . 6 ⊢ 𝑥 ∈ V
49	48	elpr 4672	. . . . 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 4014	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌})
53	13	isconn2 23443	. 2 ⊢ (𝐾 ∈ Conn ↔ (𝐾 ∈ Top ∧ (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌}))
54	2, 52, 53	sylanbrc 582	1 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {cpr 4650  ∪ cuni 4931  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448  Topctop 22920  Clsdccld 23045   Cn ccn 23253  Conncconn 23440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-top 22921  df-topon 22938  df-cld 23048  df-cn 23256  df-conn 23441

This theorem is referenced by:  connima  23454  conncn  23455  qtopconn  23738  connhmph  23818  ivthALT  36301