Theorem conima 20108

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 2400	. . . . . . 7 |- U. K = U. K
5	3, 4	cnf 19930	. . . . . 6 |- ( F e. ( J Cn K ) -> F : X --> U. K )
6	2, 5	syl 17	. . . . 5 |- ( ph -> F : X --> U. K )
7		ffun 5670	. . . . 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 5672	. . . . . 6 |- ( F : X --> U. K -> dom F = X )
11	6, 10	syl 17	. . . . 5 |- ( ph -> dom F = X )
12	9, 11	sseqtr4d 3476	. . . 4 |- ( ph -> A C_ dom F )
13		fores 5741	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
14	8, 12, 13	syl2anc 659	. . 3 |- ( ph -> ( F |` A ) : A -onto-> ( F " A ) )
15		cntop2 19925	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . 5 |- ( ph -> K e. Top )
17		imassrn 5287	. . . . . 6 |- ( F " A ) C_ ran F
18		frn 5674	. . . . . . 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 3450	. . . . 5 |- ( ph -> ( F " A ) C_ U. K )
21	4	restuni 19846	. . . . 5 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
22	16, 20, 21	syl2anc 659	. . . 4 |- ( ph -> ( F " A ) = U. ( K ↾t ( F " A ) ) )
23		foeq3 5730	. . . 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 210	. 2 |- ( ph -> ( F |` A ) : A -onto-> U. ( K ↾t ( F " A ) ) )
26	3	cnrest 19969	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
27	2, 9, 26	syl2anc 659	. . 3 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
28	4	toptopon 19616	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
29	16, 28	sylib 196	. . . 4 |- ( ph -> K e. (TopOn ` U. K ) )
30		df-ima 4953	. . . . 5 |- ( F " A ) = ran ( F |` A )
31		eqimss2 3492	. . . . 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 19970	. . . 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 1228	. . 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 2400	. . 3 |- U. ( K ↾t ( F " A ) ) = U. ( K ↾t ( F " A ) )
37	36	cnconn 20105	. 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 1228	1 |- ( ph -> ( K ↾t ( F " A ) ) e. Con )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1403   e. wcel 1840   C_ wss 3411   U.cuni 4188   dom cdm 4940   ran crn 4941   |` cres 4942   "cima 4943   Fun wfun 5517   -->wf 5519   -onto->wfo 5521   ` cfv 5523  (class class class)co 6232   ↾_t crest 14925   Topctop 19576  TopOnctopon 19577   Cn ccn 19908   Conccon 20094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-fin 7476  df-fi 7823  df-rest 14927  df-topgen 14948  df-top 19581  df-bases 19583  df-topon 19584  df-cld 19702  df-cn 19911  df-con 20095

This theorem is referenced by:  tgpconcompeqg  20792  tgpconcomp  20793