Theorem conima 20452

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 2453	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 20274	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 17	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 5736	. . . . 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 5738	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 17	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3471	. . . 4 |- ( ph -> A C_ dom F )
13		fores 5807	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
14	8, 12, 13	syl2anc 667	. . 3 |- ( ph -> ( F |` A ) : A -onto-> ( F " A ) )
15		cntop2 20269	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5182	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 5740	. . . . . . 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 3445	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 20190	. . . . 5 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
22	16, 20, 21	syl2anc 667	. . . 4 |- ( ph -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
23		foeq3 5796	. . . 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 214	. 2 |- ( ph -> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) )
26	3	cnrest 20313	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
27	2, 9, 26	syl2anc 667	. . 3 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
28	4	toptopon 19960	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 200	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 4850	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3487	. . . . 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 20314	. . . 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 1269	. . 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 214	. 2 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t ( F " A ) ) ) )
36		eqid 2453	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 20449	. 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 1269	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Con )

Syntax hints:   -> wi 4   <-> wb 188   = wceq 1446   e. wcel 1889   C_ wss 3406   U.cuni 4201   dom cdm 4837   ran crn 4838   |` cres 4839   "cima 4840   Fun wfun 5579   -->wf 5581   -onto->wfo 5583   ` cfv 5585  (class class class)co 6295   ↾_t crest 15331   Topctop 19929  TopOnctopon 19930   Cn ccn 20252   Conccon 20438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-fin 7578  df-fi 7930  df-rest 15333  df-topgen 15354  df-top 19933  df-bases 19934  df-topon 19935  df-cld 20046  df-cn 20255  df-con 20439

This theorem is referenced by:  tgpconcompeqg  21138  tgpconcomp  21139