Theorem conncn 23319

Description: A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.)

Hypotheses

Ref	Expression
conncn.x	⊢ 𝑋 = ∪ 𝐽
conncn.j	⊢ (𝜑 → 𝐽 ∈ Conn)
conncn.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
conncn.u	⊢ (𝜑 → 𝑈 ∈ 𝐾)
conncn.c	⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾))
conncn.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
conncn.1	⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈)

Assertion

Ref	Expression
conncn	⊢ (𝜑 → 𝐹:𝑋⟶𝑈)

Proof of Theorem conncn

Step	Hyp	Ref	Expression
1		conncn.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		conncn.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2730	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23139	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6	5	ffnd 6691	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7	5	frnd 6698	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
8		conncn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn)
9		dffn4 6780	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
10	6, 9	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
11		cntop2 23134	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
12	1, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
13	3	restuni 23055	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
14	12, 7, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
15		foeq3 6772	. . . . . 6 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
17	10, 16	mpbid 232	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))
18		toptopon2 22811	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	12, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3972	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 23179	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
22	19, 20, 7, 21	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
23	1, 22	mpbid 232	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
24		eqid 2730	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cnconn 23315	. . . 4 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) → (𝐾 ↾t ran 𝐹) ∈ Conn)
26	8, 17, 23, 25	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Conn)
27		conncn.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐾)
28		conncn.1	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈)
29		conncn.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
30		fnfvelrn 7054	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
31	6, 29, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
32		inelcm 4430	. . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
33	28, 31, 32	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
34		conncn.c	. . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾))
35	3, 7, 26, 27, 33, 34	connsubclo 23317	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
36		df-f 6517	. 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈))
37	6, 35, 36	sylanbrc 583	1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∩ cin 3915   ⊆ wss 3916  ∅c0 4298  ∪ cuni 4873  ran crn 5641   Fn wfn 6508  ⟶wf 6509  –onto→wfo 6511  ‘cfv 6513  (class class class)co 7389   ↾_t crest 17389  Topctop 22786  TopOnctopon 22803  Clsdccld 22909   Cn ccn 23117  Conncconn 23304

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-map 8803  df-en 8921  df-fin 8924  df-fi 9368  df-rest 17391  df-topgen 17412  df-top 22787  df-topon 22804  df-bases 22839  df-cld 22912  df-cn 23120  df-conn 23305

This theorem is referenced by:  pconnconn  35218  cvmliftmolem1  35268  cvmlift2lem9  35298  cvmlift3lem6  35311