Theorem conima 20517

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
conima.x	|- X = U. J
conima.f	|- ( ph -> F e. ( J Cn K ) )
conima.a	|- ( ph -> A C_ X )
conima.c	|- ( ph -> ( J ↾t A ) e. Con )

Assertion

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

Proof of Theorem conima

Step	Hyp	Ref	Expression
1		conima.c	. 2 |- ( ph -> ( J ↾t A ) e. Con )
2		conima.f	. . . . . 6 |- ( ph -> F e. ( J Cn K ) )
3		conima.x	. . . . . . 7 |- X = U. J
4		eqid 2471	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 20339	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 17	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 5742	. . . . 5 |- ( F : X --> U. K -> Fun F )
8	6, 7	syl 17	. . . 4 |- ( ph -> Fun F )
9		conima.a	. . . . 5 |- ( ph -> A C_ X )
10		fdm 5745	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 17	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3455	. . . 4 |- ( ph -> A C_ dom F )
13		fores 5815	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
14	8, 12, 13	syl2anc 673	. . 3 |- ( ph -> ( F |` A ) : A -onto-> ( F " A ) )
15		cntop2 20334	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5185	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 5747	. . . . . . 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 3429	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 20255	. . . . 5 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
22	16, 20, 21	syl2anc 673	. . . 4 |- ( ph -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
23		foeq3 5804	. . . 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 215	. 2 |- ( ph -> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) )
26	3	cnrest 20378	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
27	2, 9, 26	syl2anc 673	. . 3 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
28	4	toptopon 20025	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 201	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 4852	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3471	. . . . 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 20379	. . . 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 1292	. . 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 215	. 2 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) )
36		eqid 2471	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 20514	. 2 |- ( ( ( J ↾t A ) e. Con /\ ( 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. Con )
38	1, 25, 35, 37	syl3anc 1292	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Con )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   C_ wss 3390   U.cuni 4190   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   ↾_t crest 15397   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   Conccon 20503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-cn 20320  df-con 20504

This theorem is referenced by:  tgpconcompeqg  21204  tgpconcomp  21205