Theorem conncn 21229

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 = U. J
conncn.j	|- ( ph -> J e. Conn )
conncn.f	|- ( ph -> F e. ( J Cn K ) )
conncn.u	|- ( ph -> U e. K )
conncn.c	|- ( ph -> U e. ( Clsd ` K ) )
conncn.a	|- ( ph -> A e. X )
conncn.1	|- ( ph -> ( F ` A ) e. U )

Assertion

Ref	Expression
conncn	|- ( ph -> F : X --> U )

Proof of Theorem conncn

Step	Hyp	Ref	Expression
1		conncn.f	. . . 4 |- ( ph -> F e. ( J Cn K ) )
2		conncn.x	. . . . 5 |- X = U. J
3		eqid 2622	. . . . 5 |- U. K = U. K
4	2, 3	cnf 21050	. . . 4 |- ( F e. ( J Cn K ) -> F : X --> U. K )
5	1, 4	syl 17	. . 3 |- ( ph -> F : X --> U. K )
6		ffn 6045	. . 3 |- ( F : X --> U. K -> F Fn X )
7	5, 6	syl 17	. 2 |- ( ph -> F Fn X )
8		frn 6053	. . . 4 |- ( F : X --> U. K -> ran F C_ U. K )
9	5, 8	syl 17	. . 3 |- ( ph -> ran F C_ U. K )
10		conncn.j	. . . 4 |- ( ph -> J e. Conn )
11		dffn4 6121	. . . . . 6 |- ( F Fn X <-> F : X -onto-> ran F )
12	7, 11	sylib 208	. . . . 5 |- ( ph -> F : X -onto-> ran F )
13		cntop2 21045	. . . . . . . 8 |- ( F e. ( J Cn K ) -> K e. Top )
14	1, 13	syl 17	. . . . . . 7 |- ( ph -> K e. Top )
15	3	restuni 20966	. . . . . . 7 |- ( ( K e. Top /\ ran F C_ U. K ) -> ran F = U. ( K ↾t ran F ) )
16	14, 9, 15	syl2anc 693	. . . . . 6 |- ( ph -> ran F = U. ( K ↾t ran F ) )
17		foeq3 6113	. . . . . 6 |- ( ran F = U. ( K ↾t ran F ) -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( K ↾t ran F ) ) )
18	16, 17	syl 17	. . . . 5 |- ( ph -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( K ↾t ran F ) ) )
19	12, 18	mpbid 222	. . . 4 |- ( ph -> F : X -onto-> U. ( K ↾t ran F ) )
20	3	toptopon 20722	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
21	14, 20	sylib 208	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
22		ssid 3624	. . . . . . 7 |- ran F C_ ran F
23	22	a1i 11	. . . . . 6 |- ( ph -> ran F C_ ran F )
24		cnrest2 21090	. . . . . 6 |- ( ( K e. (TopOn ` U. K ) /\ ran F C_ ran F /\ ran F C_ U. K ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t ran F ) ) ) )
25	21, 23, 9, 24	syl3anc 1326	. . . . 5 |- ( ph -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t ran F ) ) ) )
26	1, 25	mpbid 222	. . . 4 |- ( ph -> F e. ( J Cn ( K ↾t ran F ) ) )
27		eqid 2622	. . . . 5 |- U. ( K ↾t ran F ) = U. ( K ↾t ran F )
28	27	cnconn 21225	. . . 4 |- ( ( J e. Conn /\ F : X -onto-> U. ( K ↾t ran F ) /\ F e. ( J Cn ( K ↾t ran F ) ) ) -> ( K ↾t ran F ) e. Conn )
29	10, 19, 26, 28	syl3anc 1326	. . 3 |- ( ph -> ( K ↾t ran F ) e. Conn )
30		conncn.u	. . 3 |- ( ph -> U e. K )
31		conncn.1	. . . 4 |- ( ph -> ( F ` A ) e. U )
32		conncn.a	. . . . 5 |- ( ph -> A e. X )
33		fnfvelrn 6356	. . . . 5 |- ( ( F Fn X /\ A e. X ) -> ( F ` A ) e. ran F )
34	7, 32, 33	syl2anc 693	. . . 4 |- ( ph -> ( F ` A ) e. ran F )
35		inelcm 4032	. . . 4 |- ( ( ( F ` A ) e. U /\ ( F ` A ) e. ran F ) -> ( U i^i ran F ) =/= (/) )
36	31, 34, 35	syl2anc 693	. . 3 |- ( ph -> ( U i^i ran F ) =/= (/) )
37		conncn.c	. . 3 |- ( ph -> U e. ( Clsd ` K ) )
38	3, 9, 29, 30, 36, 37	connsubclo 21227	. 2 |- ( ph -> ran F C_ U )
39		df-f 5892	. 2 |- ( F : X --> U <-> ( F Fn X /\ ran F C_ U ) )
40	7, 38, 39	sylanbrc 698	1 |- ( ph -> F : X --> U )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   Cn ccn 21028  Conncconn 21214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cn 21031  df-conn 21215

This theorem is referenced by:  pconnconn  31213  cvmliftmolem1  31263  cvmlift2lem9  31293  cvmlift3lem6  31306