Theorem conncn 22649

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 2737	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 22469	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6	5	ffnd 6638	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7	5	frnd 6645	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
8		conncn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn)
9		dffn4 6731	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
10	6, 9	sylib 217	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
11		cntop2 22464	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
12	1, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
13	3	restuni 22385	. . . . . . 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 6723	. . . . . 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 22139	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	12, 18	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3954	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 22509	. . . . . 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 2737	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cnconn 22645	. . . 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 6997	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
31	6, 29, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
32		inelcm 4409	. . . 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 22647	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
36		df-f 6469	. 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 1540   ∈ wcel 2105   ≠ wne 2941   ∩ cin 3896   ⊆ wss 3897  ∅c0 4267  ∪ cuni 4850  ran crn 5608   Fn wfn 6460  ⟶wf 6461  –onto→wfo 6463  ‘cfv 6465  (class class class)co 7315   ↾_t crest 17201  Topctop 22114  TopOnctopon 22131  Clsdccld 22239   Cn ccn 22447  Conncconn 22634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-map 8665  df-en 8782  df-fin 8785  df-fi 9240  df-rest 17203  df-topgen 17224  df-top 22115  df-topon 22132  df-bases 22168  df-cld 22242  df-cn 22450  df-conn 22635

This theorem is referenced by:  pconnconn  33298  cvmliftmolem1  33348  cvmlift2lem9  33378  cvmlift3lem6  33391