MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conima Structured version   Visualization version   Unicode version

Theorem conima 20517
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 2471 . . . . . . 7  |-  U. K  =  U. K
53, 4cnf 20339 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
62, 5syl 17 . . . . 5  |-  ( ph  ->  F : X --> U. K
)
7 ffun 5742 . . . . 5  |-  ( F : X --> U. K  ->  Fun  F )
86, 7syl 17 . . . 4  |-  ( ph  ->  Fun  F )
9 conima.a . . . . 5  |-  ( ph  ->  A  C_  X )
10 fdm 5745 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
116, 10syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  X )
129, 11sseqtr4d 3455 . . . 4  |-  ( ph  ->  A  C_  dom  F )
13 fores 5815 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
148, 12, 13syl2anc 673 . . 3  |-  ( ph  ->  ( F  |`  A ) : A -onto-> ( F
" A ) )
15 cntop2 20334 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 17 . . . . 5  |-  ( ph  ->  K  e.  Top )
17 imassrn 5185 . . . . . 6  |-  ( F
" A )  C_  ran  F
18 frn 5747 . . . . . . 7  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
196, 18syl 17 . . . . . 6  |-  ( ph  ->  ran  F  C_  U. K
)
2017, 19syl5ss 3429 . . . . 5  |-  ( ph  ->  ( F " A
)  C_  U. K )
214restuni 20255 . . . . 5  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( F " A
)  =  U. ( Kt  ( F " A ) ) )
2216, 20, 21syl2anc 673 . . . 4  |-  ( ph  ->  ( F " A
)  =  U. ( Kt  ( F " A ) ) )
23 foeq3 5804 . . . 4  |-  ( ( F " A )  =  U. ( Kt  ( F " A ) )  ->  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2422, 23syl 17 . . 3  |-  ( ph  ->  ( ( F  |`  A ) : A -onto->
( F " A
)  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2514, 24mpbid 215 . 2  |-  ( ph  ->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) )
263cnrest 20378 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
272, 9, 26syl2anc 673 . . 3  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
284toptopon 20025 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2916, 28sylib 201 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
30 df-ima 4852 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
31 eqimss2 3471 . . . . 5  |-  ( ( F " A )  =  ran  ( F  |`  A )  ->  ran  ( F  |`  A ) 
C_  ( F " A ) )
3230, 31mp1i 13 . . . 4  |-  ( ph  ->  ran  ( F  |`  A )  C_  ( F " A ) )
33 cnrest2 20379 . . . 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 1292 . . 3  |-  ( ph  ->  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
)  <->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) ) )
3527, 34mpbid 215 . 2  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) )
36 eqid 2471 . . 3  |-  U. ( Kt  ( F " A ) )  =  U. ( Kt  ( F " A ) )
3736cnconn 20514 . 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 1292 1  |-  ( ph  ->  ( Kt  ( F " A ) )  e. 
Con )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904    C_ wss 3390   U.cuni 4190   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995    Cn ccn 20317   Conccon 20503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-cn 20320  df-con 20504
This theorem is referenced by:  tgpconcompeqg  21204  tgpconcomp  21205
  Copyright terms: Public domain W3C validator