Theorem conima 19685

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 2460	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 19506	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 16	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 5724	. . . . 5 |- ( F : X --> U. K -> Fun F )
8	6, 7	syl 16	. . . 4 |- ( ph -> Fun F )
9		conima.a	. . . . 5 |- ( ph -> A C_ X )
10		fdm 5726	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 16	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3534	. . . 4 |- ( ph -> A C_ dom F )
13		fores 5795	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
14	8, 12, 13	syl2anc 661	. . 3 |- ( ph -> ( F |` A ) : A -onto-> ( F " A ) )
15		cntop2 19501	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 16	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5339	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 5728	. . . . . . 7 |- ( F : X --> U. K -> ran F C_ U. K )
19	6, 18	syl 16	. . . . . 6 |- ( ph -> ran F C_ U. K )
20	17, 19	syl5ss 3508	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 19422	. . . . 5 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
22	16, 20, 21	syl2anc 661	. . . 4 |- ( ph -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
23		foeq3 5784	. . . 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 16	. . 3 |- ( ph -> ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) ) )
25	14, 24	mpbid 210	. 2 |- ( ph -> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) )
26	3	cnrest 19545	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
27	2, 9, 26	syl2anc 661	. . 3 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
28	4	toptopon 19194	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 196	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 5005	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3550	. . . . 5 |- ( ( F " A ) = ran ( F |` A ) -> ran ( F |` A ) C_ ( F " A ) )
32	30, 31	mp1i 12	. . . 4 |- ( ph -> ran ( F |` A ) C_ ( F " A ) )
33		cnrest2 19546	. . . 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 1223	. . 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 210	. 2 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) )
36		eqid 2460	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 19682	. 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 1223	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Con )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1374   e. wcel 1762   C_ wss 3469   U.cuni 4238   dom cdm 4992   ran crn 4993   |` cres 4994   "cima 4995   Fun wfun 5573   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   ↾_t crest 14665   Topctop 19154  TopOnctopon 19155   Cn ccn 19484   Conccon 19671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-fin 7510  df-fi 7860  df-rest 14667  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cld 19279  df-cn 19487  df-con 19672

This theorem is referenced by:  tgpconcompeqg  20338  tgpconcomp  20339