Theorem concn 21039

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

Assertion

Ref	Expression
concn	⊢ (𝜑 → 𝐹:𝑋⟶𝑈)

Proof of Theorem concn

Step	Hyp	Ref	Expression
1		concn.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		concn.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3		eqid 2610	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 20860	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6		ffn 5958	. . 3 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
8		frn 5966	. . . 4 ⊢ (𝐹:𝑋⟶∪ 𝐾 → ran 𝐹 ⊆ ∪ 𝐾)
9	5, 8	syl 17	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
10		concn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Con)
11		dffn4 6034	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
12	7, 11	sylib 207	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
13		cntop2 20855	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
14	1, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
15	3	restuni 20776	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
16	14, 9, 15	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
17		foeq3 6026	. . . . . 6 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
19	12, 18	mpbid 221	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))
20	3	toptopon 20548	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
21	14, 20	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
22		ssid 3587	. . . . . . 7 ⊢ ran 𝐹 ⊆ ran 𝐹
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
24		cnrest2 20900	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
25	21, 23, 9, 24	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
26	1, 25	mpbid 221	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
27		eqid 2610	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
28	27	cnconn 21035	. . . 4 ⊢ ((𝐽 ∈ Con ∧ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) → (𝐾 ↾t ran 𝐹) ∈ Con)
29	10, 19, 26, 28	syl3anc 1318	. . 3 ⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Con)
30		concn.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐾)
31		concn.1	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈)
32		concn.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
33		fnfvelrn 6264	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
34	7, 32, 33	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
35		inelcm 3984	. . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
36	31, 34, 35	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
37		concn.c	. . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾))
38	3, 9, 29, 30, 36, 37	consubclo 21037	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
39		df-f 5808	. 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈))
40	7, 38, 39	sylanbrc 695	1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838  Conccon 21024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cn 20841  df-con 21025

This theorem is referenced by:  pconcon  30467  cvmliftmolem1  30517  cvmlift2lem9  30547  cvmlift3lem6  30560