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Theorem conima 17167
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 2296 . . . . . . 7  |-  U. K  =  U. K
53, 4cnf 16992 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
62, 5syl 15 . . . . 5  |-  ( ph  ->  F : X --> U. K
)
7 ffun 5407 . . . . 5  |-  ( F : X --> U. K  ->  Fun  F )
86, 7syl 15 . . . 4  |-  ( ph  ->  Fun  F )
9 conima.a . . . . 5  |-  ( ph  ->  A  C_  X )
10 fdm 5409 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
116, 10syl 15 . . . . 5  |-  ( ph  ->  dom  F  =  X )
129, 11sseqtr4d 3228 . . . 4  |-  ( ph  ->  A  C_  dom  F )
13 fores 5476 . . . 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 16987 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 15 . . . . 5  |-  ( ph  ->  K  e.  Top )
17 imassrn 5041 . . . . . 6  |-  ( F
" A )  C_  ran  F
18 frn 5411 . . . . . . 7  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
196, 18syl 15 . . . . . 6  |-  ( ph  ->  ran  F  C_  U. K
)
2017, 19syl5ss 3203 . . . . 5  |-  ( ph  ->  ( F " A
)  C_  U. K )
214restuni 16909 . . . . 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 5465 . . . 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 17029 . . . 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 16687 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2916, 28sylib 188 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
30 df-ima 4718 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
31 eqimss2 3244 . . . . 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 17030 . . . 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 2296 . . 3  |-  U. ( Kt  ( F " A ) )  =  U. ( Kt  ( F " A ) )
3736cnconn 17164 . 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 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648    Cn ccn 16970   Conccon 17153
This theorem is referenced by:  tgpconcompeqg  17810  tgpconcomp  17811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-cn 16973  df-con 17154
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