Theorem conncn 22577

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 2738	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 22397	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6	5	ffnd 6601	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7	5	frnd 6608	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
8		conncn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn)
9		dffn4 6694	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
10	6, 9	sylib 217	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
11		cntop2 22392	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
12	1, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
13	3	restuni 22313	. . . . . . 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 6686	. . . . . 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 231	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))
18		toptopon2 22067	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	12, 18	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3944	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 22437	. . . . . 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 1370	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
23	1, 22	mpbid 231	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
24		eqid 2738	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cnconn 22573	. . . 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 1370	. . 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 6958	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
31	6, 29, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
32		inelcm 4398	. . . 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 22575	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
36		df-f 6437	. 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 205   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ran crn 5590   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  TopOnctopon 22059  Clsdccld 22167   Cn ccn 22375  Conncconn 22562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8617  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-cn 22378  df-conn 22563

This theorem is referenced by:  pconnconn  33193  cvmliftmolem1  33243  cvmlift2lem9  33273  cvmlift3lem6  33286