Theorem conncn 23370

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 2736	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23190	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
6	5	ffnd 6663	. 2 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7	5	frnd 6670	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
8		conncn.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Conn)
9		dffn4 6752	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
10	6, 9	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
11		cntop2 23185	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
12	1, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
13	3	restuni 23106	. . . . . . 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 6744	. . . . . 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 22862	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	12, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3957	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 23230	. . . . . 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 2736	. . . . 5 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cnconn 23366	. . . 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 7025	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
31	6, 29, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
32		inelcm 4417	. . . 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 23368	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑈)
36		df-f 6496	. 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 1541   ∈ wcel 2113   ≠ wne 2932   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ∪ cuni 4863  ran crn 5625   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ↾_t crest 17340  Topctop 22837  TopOnctopon 22854  Clsdccld 22960   Cn ccn 23168  Conncconn 23355

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-map 8765  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17342  df-topgen 17363  df-top 22838  df-topon 22855  df-bases 22890  df-cld 22963  df-cn 23171  df-conn 23356

This theorem is referenced by:  pconnconn  35425  cvmliftmolem1  35475  cvmlift2lem9  35505  cvmlift3lem6  35518