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Theorem conima 17151
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  ->  ( Jt  A )  e.  Con )
Assertion
Ref Expression
conima  |-  ( ph  ->  ( Kt  ( F " A ) )  e. 
Con )

Proof of Theorem conima
StepHypRef Expression
1 conima.c . 2  |-  ( ph  ->  ( Jt  A )  e.  Con )
2 conima.f . . . . . 6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 conima.x . . . . . . 7  |-  X  = 
U. J
4 eqid 2283 . . . . . . 7  |-  U. K  =  U. K
53, 4cnf 16976 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
62, 5syl 15 . . . . 5  |-  ( ph  ->  F : X --> U. K
)
7 ffun 5391 . . . . 5  |-  ( F : X --> U. K  ->  Fun  F )
86, 7syl 15 . . . 4  |-  ( ph  ->  Fun  F )
9 conima.a . . . . 5  |-  ( ph  ->  A  C_  X )
10 fdm 5393 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
116, 10syl 15 . . . . 5  |-  ( ph  ->  dom  F  =  X )
129, 11sseqtr4d 3215 . . . 4  |-  ( ph  ->  A  C_  dom  F )
13 fores 5460 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
148, 12, 13syl2anc 642 . . 3  |-  ( ph  ->  ( F  |`  A ) : A -onto-> ( F
" A ) )
15 cntop2 16971 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 15 . . . . 5  |-  ( ph  ->  K  e.  Top )
17 imassrn 5025 . . . . . 6  |-  ( F
" A )  C_  ran  F
18 frn 5395 . . . . . . 7  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
196, 18syl 15 . . . . . 6  |-  ( ph  ->  ran  F  C_  U. K
)
2017, 19syl5ss 3190 . . . . 5  |-  ( ph  ->  ( F " A
)  C_  U. K )
214restuni 16893 . . . . 5  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( F " A
)  =  U. ( Kt  ( F " A ) ) )
2216, 20, 21syl2anc 642 . . . 4  |-  ( ph  ->  ( F " A
)  =  U. ( Kt  ( F " A ) ) )
23 foeq3 5449 . . . 4  |-  ( ( F " A )  =  U. ( Kt  ( F " A ) )  ->  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2422, 23syl 15 . . 3  |-  ( ph  ->  ( ( F  |`  A ) : A -onto->
( F " A
)  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2514, 24mpbid 201 . 2  |-  ( ph  ->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) )
263cnrest 17013 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
272, 9, 26syl2anc 642 . . 3  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
284toptopon 16671 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2916, 28sylib 188 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
30 df-ima 4702 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
31 eqimss2 3231 . . . . 5  |-  ( ( F " A )  =  ran  ( F  |`  A )  ->  ran  ( F  |`  A ) 
C_  ( F " A ) )
3230, 31mp1i 11 . . . 4  |-  ( ph  ->  ran  ( F  |`  A )  C_  ( F " A ) )
33 cnrest2 17014 . . . 4  |-  ( ( K  e.  (TopOn `  U. K )  /\  ran  ( F  |`  A ) 
C_  ( F " A )  /\  ( F " A )  C_  U. K )  ->  (
( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  <-> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) ) )
3429, 32, 20, 33syl3anc 1182 . . 3  |-  ( ph  ->  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
)  <->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) ) )
3527, 34mpbid 201 . 2  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) )
36 eqid 2283 . . 3  |-  U. ( Kt  ( F " A ) )  =  U. ( Kt  ( F " A ) )
3736cnconn 17148 . 2  |-  ( ( ( Jt  A )  e.  Con  /\  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) )  /\  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) )  -> 
( Kt  ( F " A ) )  e. 
Con )
381, 25, 35, 37syl3anc 1182 1  |-  ( ph  ->  ( Kt  ( F " A ) )  e. 
Con )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632    Cn ccn 16954   Conccon 17137
This theorem is referenced by:  tgpconcompeqg  17794  tgpconcomp  17795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-cn 16957  df-con 17138
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