Theorem connima 21228

Description: The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
connima.x	|- X = U. J
connima.f	|- ( ph -> F e. ( J Cn K ) )
connima.a	|- ( ph -> A C_ X )
connima.c	|- ( ph -> ( J ↾t A ) e. Conn )

Assertion

Ref	Expression
connima	|- ( ph -> ( K ↾t ( F " A ) ) e. Conn )

Proof of Theorem connima

Step	Hyp	Ref	Expression
1		connima.c	. 2 |- ( ph -> ( J ↾t A ) e. Conn )
2		connima.f	. . . . . 6 |- ( ph -> F e. ( J Cn K ) )
3		connima.x	. . . . . . 7 |- X = U. J
4		eqid 2622	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 21050	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 17	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 6048	. . . . 5 |- ( F : X --> U. K -> Fun F )
8	6, 7	syl 17	. . . 4 |- ( ph -> Fun F )
9		connima.a	. . . . 5 |- ( ph -> A C_ X )
10		fdm 6051	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 17	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3642	. . . 4 |- ( ph -> A C_ dom F )
13		fores 6124	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
14	8, 12, 13	syl2anc 693	. . 3 |- ( ph -> ( F |` A ) : A -onto-> ( F " A ) )
15		cntop2 21045	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5477	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 6053	. . . . . . 7 |- ( F : X --> U. K -> ran F C_ U. K )
19	6, 18	syl 17	. . . . . 6 |- ( ph -> ran F C_ U. K )
20	17, 19	syl5ss 3614	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 20966	. . . . 5 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
22	16, 20, 21	syl2anc 693	. . . 4 |- ( ph -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
23		foeq3 6113	. . . 4 |- ( ( F " A ) = U. ( K ↾t ( F " A ) ) -> ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) ) )
24	22, 23	syl 17	. . 3 |- ( ph -> ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) ) )
25	14, 24	mpbid 222	. 2 |- ( ph -> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) )
26	3	cnrest 21089	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
27	2, 9, 26	syl2anc 693	. . 3 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
28	4	toptopon 20722	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 208	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 5127	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3658	. . . . 5 |- ( ( F " A ) = ran ( F |` A ) -> ran ( F |` A ) C_ ( F " A ) )
32	30, 31	mp1i 13	. . . 4 |- ( ph -> ran ( F |` A ) C_ ( F " A ) )
33		cnrest2 21090	. . . 4 |- ( ( K e. (TopOn ` U. K ) /\ ran ( F |` A ) C_ ( F " A ) /\ ( F " A ) C_ U. K ) -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) ) )
34	29, 32, 20, 33	syl3anc 1326	. . 3 |- ( ph -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) ) )
35	27, 34	mpbid 222	. 2 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) )
36		eqid 2622	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 21225	. 2 |- ( ( ( J ↾t A ) e. Conn /\ ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) /\ ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) ) -> ( K ↾t ( F " A ) ) e. Conn )
38	1, 25, 35, 37	syl3anc 1326	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Conn )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   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:  tgpconncompeqg  21915  tgpconncomp  21916