Theorem conncn 23737

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 2761	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23557	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6	5	ffnd 6708	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7	5	frnd 6716	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
8		conncn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn)
9		dffn4 6800	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
10	6, 9	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
11		cntop2 23552	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
12	1, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
13	3	restuni 23473	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
14	12, 7, 13	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
15		foeq3 6792	. . . . . 6 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
16	14, 15	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹)))
17	10, 16	mpbid 235	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→∪ (𝐾 ↾t ran 𝐹))
18		toptopon2 23229	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	12, 18	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3954	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 23597	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
23	1, 22	mpbid 235	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
24		eqid 2761	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cnconn 23733	. . . 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 1398	. . 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 7078	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
31	6, 29, 30	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
32		inelcm 4418	. . . 4 ⊢ (((𝐹‘𝐴) ∈ 𝑈 ∧ (𝐹‘𝐴) ∈ ran 𝐹) → (𝑈 ∩ ran 𝐹) ≠ ∅)
33	28, 31, 32	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑈 ∩ ran 𝐹) ≠ ∅)
34		conncn.c	. . 3 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾))
35	3, 7, 26, 27, 33, 34	connsubclo 23735	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
36		df-f 6541	. 2 ⊢ (𝐹:𝑋⟶𝑈 ↔ (𝐹 Fn 𝑋 ∧ ran 𝐹 ⊆ 𝑈))
37	6, 35, 36	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:𝑋⟶𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ran crn 5652   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7418   ↾_t crest 17584  Topctop 23204  TopOnctopon 23221  Clsdccld 23327   Cn ccn 23535  Conncconn 23722

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-map 8842  df-en 8967  df-fin 8970  df-fi 9396  df-rest 17586  df-topgen 17607  df-top 23205  df-topon 23222  df-bases 23257  df-cld 23330  df-cn 23538  df-conn 23723

This theorem is used by:  pconnconn  35975  cvmliftmolem1  36025  cvmlift2lem9  36055  cvmlift3lem6  36068