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Theorem conima 19685
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 2460 . . . . . . 7  |-  U. K  =  U. K
53, 4cnf 19506 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
62, 5syl 16 . . . . 5  |-  ( ph  ->  F : X --> U. K
)
7 ffun 5724 . . . . 5  |-  ( F : X --> U. K  ->  Fun  F )
86, 7syl 16 . . . 4  |-  ( ph  ->  Fun  F )
9 conima.a . . . . 5  |-  ( ph  ->  A  C_  X )
10 fdm 5726 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
116, 10syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  X )
129, 11sseqtr4d 3534 . . . 4  |-  ( ph  ->  A  C_  dom  F )
13 fores 5795 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
148, 12, 13syl2anc 661 . . 3  |-  ( ph  ->  ( F  |`  A ) : A -onto-> ( F
" A ) )
15 cntop2 19501 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . 5  |-  ( ph  ->  K  e.  Top )
17 imassrn 5339 . . . . . 6  |-  ( F
" A )  C_  ran  F
18 frn 5728 . . . . . . 7  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
196, 18syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  U. K
)
2017, 19syl5ss 3508 . . . . 5  |-  ( ph  ->  ( F " A
)  C_  U. K )
214restuni 19422 . . . . 5  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( F " A
)  =  U. ( Kt  ( F " A ) ) )
2216, 20, 21syl2anc 661 . . . 4  |-  ( ph  ->  ( F " A
)  =  U. ( Kt  ( F " A ) ) )
23 foeq3 5784 . . . 4  |-  ( ( F " A )  =  U. ( Kt  ( F " A ) )  ->  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2422, 23syl 16 . . 3  |-  ( ph  ->  ( ( F  |`  A ) : A -onto->
( F " A
)  <->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) ) )
2514, 24mpbid 210 . 2  |-  ( ph  ->  ( F  |`  A ) : A -onto-> U. ( Kt  ( F " A ) ) )
263cnrest 19545 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
272, 9, 26syl2anc 661 . . 3  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
284toptopon 19194 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2916, 28sylib 196 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
30 df-ima 5005 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
31 eqimss2 3550 . . . . 5  |-  ( ( F " A )  =  ran  ( F  |`  A )  ->  ran  ( F  |`  A ) 
C_  ( F " A ) )
3230, 31mp1i 12 . . . 4  |-  ( ph  ->  ran  ( F  |`  A )  C_  ( F " A ) )
33 cnrest2 19546 . . . 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 1223 . . 3  |-  ( ph  ->  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
)  <->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) ) )
3527, 34mpbid 210 . 2  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  ( F " A ) ) ) )
36 eqid 2460 . . 3  |-  U. ( Kt  ( F " A ) )  =  U. ( Kt  ( F " A ) )
3736cnconn 19682 . 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 1223 1  |-  ( ph  ->  ( Kt  ( F " A ) )  e. 
Con )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762    C_ wss 3469   U.cuni 4238   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   ↾t crest 14665   Topctop 19154  TopOnctopon 19155    Cn ccn 19484   Conccon 19671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-fin 7510  df-fi 7860  df-rest 14667  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cld 19279  df-cn 19487  df-con 19672
This theorem is referenced by:  tgpconcompeqg  20338  tgpconcomp  20339
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