Theorem cnconn 23767

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⟶onto𝑌 ∧ 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 23586	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		df-ne 2960	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
4		eqid 2763	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
5		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐽 ∈ Conn)
6		simpl3 1212	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹 ∈ (𝐽 Cn 𝐾))
7		simprl 783	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)))
8	7	elin1d 4104	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ 𝐾)
9		cnima 23610	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
10	6, 8, 9	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
11		elssuni 4902	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
12	8, 11	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ∪ 𝐾)
13		cnconn.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑌 = ∪ 𝐾
14	12, 13	sseqtrrdi 3954	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ 𝑌)
15		simpl2 1211	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝐹:𝑋⟶onto𝑌)
16		forn 6818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑋⟶onto𝑌 → ran 𝐹 = 𝑌)
17	15, 16	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → ran 𝐹 = 𝑌)
18	14, 17	sseqtrrd 3950	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ ran 𝐹)
19		df-rn 5666	. . . . . . . . . . . . . . . 16 ⊢ ran 𝐹 = dom ◡𝐹
20	18, 19	sseqtrdi 3953	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ⊆ dom ◡𝐹)
21		sseqin2 4123	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ dom ◡𝐹 ↔ (dom ◡𝐹 ∩ 𝑥) = 𝑥)
22	20, 21	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) = 𝑥)
23		simprr 785	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅)
24	22, 23	eqnetrd 3026	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
25		imadisj 6088	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹 “ 𝑥) = ∅ ↔ (dom ◡𝐹 ∩ 𝑥) = ∅)
26	25	necon3bii 3011	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑥) ≠ ∅ ↔ (dom ◡𝐹 ∩ 𝑥) ≠ ∅)
27	24, 26	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ≠ ∅)
28	7	elin2d 4105	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Clsd‘𝐾))
29		cnclima 23613	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
30	6, 28, 29	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) ∈ (Clsd‘𝐽))
31	4, 5, 10, 27, 30	connclo 23760	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = ∪ 𝐽)
32	4, 13	cnf 23591	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
33		fdm 6738	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶𝑌 → dom 𝐹 = ∪ 𝐽)
34	6, 32, 33	3syl 19	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = ∪ 𝐽)
35		fof 6815	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹:𝑋⟶𝑌)
36		fdm 6738	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
37	15, 35, 36	3syl 19	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → dom 𝐹 = 𝑋)
38	31, 34, 37	3eqtr2d 2804	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (◡𝐹 “ 𝑥) = 𝑋)
39	38	imaeq2d 6063	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ 𝑋))
40		foimacnv 6861	. . . . . . . . . 10 ⊢ ((𝐹:𝑋⟶onto𝑌 ∧ 𝑥 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
41	15, 14, 40	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
42		foima 6820	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶onto𝑌 → (𝐹 “ 𝑋) = 𝑌)
43	15, 42	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → (𝐹 “ 𝑋) = 𝑌)
44	39, 41, 43	3eqtr3d 2806	. . . . . . . 8 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) ∧ 𝑥 ≠ ∅)) → 𝑥 = 𝑌)
45	44	expr 462	. . . . . . 7 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 ≠ ∅ → 𝑥 = 𝑌))
46	3, 45	biimtrrid 246	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (¬ 𝑥 = ∅ → 𝑥 = 𝑌))
47	46	orrd 877	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → (𝑥 = ∅ ∨ 𝑥 = 𝑌))
48		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
49	48	elpr 4609	. . . . 5 ⊢ (𝑥 ∈ {∅, 𝑌} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝑌))
50	47, 49	sylibr 237	. . . 4 ⊢ (((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾))) → 𝑥 ∈ {∅, 𝑌})
51	50	ex 418	. . 3 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ (𝐾 ∩ (Clsd‘𝐾)) → 𝑥 ∈ {∅, 𝑌}))
52	51	ssrdv 3919	. 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌})
53	13	isconn2 23759	. 2 ⊢ (𝐾 ∈ Conn ↔ (𝐾 ∈ Top ∧ (𝐾 ∩ (Clsd‘𝐾)) ⊆ {∅, 𝑌}))
54	2, 52, 53	sylanbrc 595	1 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {cpr 4590  ∪ cuni 4870  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ⟶wf 6551  ⟶ontowfo 6553  ‘cfv 6555  (class class class)co 7440  Topctop 23238  Clsdccld 23361   Cn ccn 23569  Conncconn 23756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-top 23239  df-topon 23256  df-cld 23364  df-cn 23572  df-conn 23757

This theorem is used by:  connima  23770  conncn  23771  qtopconn  24055  connhmph  24135  ivthALT  37300