Theorem cnconn 23740

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 23559	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		df-ne 2956	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
4		eqid 2760	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐽 ∈ Conn)
6		simpl3 1212	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹 ∈ (𝐽 Cn 𝐾))
7		simprl 783	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)))
8	7	elin1d 4101	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ 𝐾)
9		cnima 23583	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
10	6, 8, 9	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
11		elssuni 4898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
12	8, 11	syl 18	. . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ 𝑌)
15		simpl2 1211	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹:𝑋–onto→𝑌)
16		forn 6799	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
17	15, 16	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → ran 𝐹 = 𝑌)
18	14, 17	sseqtrrd 3945	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ran 𝐹)
19		df-rn 5661	. . . . . . . . . . . . . . . 16 ⊢ ran 𝐹 = dom ◡𝐹
20	18, 19	sseqtrdi 3948	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ dom ◡𝐹)
21		sseqin2 4120	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ dom ◡𝐹 ↔ (dom ◡𝐹 ∩ 𝑥) = 𝑥)
22	20, 21	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) = 𝑥)
23		simprr 785	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅)
24	22, 23	eqnetrd 3022	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
25		imadisj 6076	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑥) = ∅ ↔ (dom ◡𝐹 ∩ 𝑥) = ∅)
26	25	necon3bii 3007	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑥) ≠ ∅ ↔ (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
27	24, 26	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ≠ ∅)
28	7	elin2d 4102	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Clsd‘𝐾))
29		cnclima 23586	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
30	6, 28, 29	syl2anc 596	. . . . . . . . . . . 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 23733	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = ∪ 𝐽)
32	4, 13	cnf 23564	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
33		fdm 6719	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶𝑌 → dom 𝐹 = ∪ 𝐽)
34	6, 32, 33	3syl 19	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = ∪ 𝐽)
35		fof 6796	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
36		fdm 6719	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
37	15, 35, 36	3syl 19	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = 𝑋)
38	31, 34, 37	3eqtr2d 2801	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = 𝑋)
39	38	imaeq2d 6051	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ 𝑋))
40		foimacnv 6842	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑥 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
41	15, 14, 40	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
42		foima 6801	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
43	15, 42	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ 𝑋) = 𝑌)
44	39, 41, 43	3eqtr3d 2803	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 = 𝑌)
45	44	expr 462	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 ≠ ∅ → 𝑥 = 𝑌))
46	3, 45	biimtrrid 246	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (¬ 𝑥 = ∅ → 𝑥 = 𝑌))
47	46	orrd 877	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 = ∅ ∨ 𝑥 = 𝑌))
48		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
49	48	elpr 4608	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝑌} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝑌))
50	47, 49	sylibr 237	. . . 4 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → 𝑥 ∈ {∅, 𝑌})
51	50	ex 418	. . 3 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) → 𝑥 ∈ {∅, 𝑌}))
52	51	ssrdv 3914	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌})
53	13	isconn2 23732	. 2 ⊢ (𝐾 ∈ Conn ↔ (𝐾 ∈ Top ∧ (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌}))
54	2, 52, 53	sylanbrc 595	1 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  {cpr 4585  ∪ cuni 4866  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420  Topctop 23211  Clsdccld 23334   Cn ccn 23542  Conncconn 23729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-top 23212  df-topon 23229  df-cld 23337  df-cn 23545  df-conn 23730

This theorem is used by:  connima  23743  conncn  23744  qtopconn  24028  connhmph  24108  ivthALT  37123