Theorem conima 19162

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 2454	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 18983	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 16	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 5670	. . . . 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 5672	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 16	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3502	. . . 4 |- ( ph -> A C_ dom F )
13		fores 5738	. . . 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 18978	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 16	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5289	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 5674	. . . . . . 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 3476	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 18899	. . . . 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 5727	. . . 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 19022	. . . 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 18671	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 196	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 4962	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3518	. . . . 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 19023	. . . 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 1219	. . 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 2454	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 19159	. 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 1219	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Con )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1370   e. wcel 1758   C_ wss 3437   U.cuni 4200   dom cdm 4949   ran crn 4950   |` cres 4951   "cima 4952   Fun wfun 5521   -->wf 5523   -onto->wfo 5525   ` cfv 5527  (class class class)co 6201   ↾_t crest 14479   Topctop 18631  TopOnctopon 18632   Cn ccn 18961   Conccon 19148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-fin 7425  df-fi 7773  df-rest 14481  df-topgen 14502  df-top 18636  df-bases 18638  df-topon 18639  df-cld 18756  df-cn 18964  df-con 19149

This theorem is referenced by:  tgpconcompeqg  19815  tgpconcomp  19816