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Theorem conima 19162
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 2454 . . . . . . 7  |-  U. K  =  U. K
53, 4cnf 18983 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
62, 5syl 16 . . . . 5  |-  ( ph  ->  F : X --> U. K
)
7 ffun 5670 . . . . 5  |-  ( F : X --> U. K  ->  Fun  F )
86, 7syl 16 . . . 4  |-  ( ph  ->  Fun  F )
9 conima.a . . . . 5  |-  ( ph  ->  A  C_  X )
10 fdm 5672 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
116, 10syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  X )
129, 11sseqtr4d 3502 . . . 4  |-  ( ph  ->  A  C_  dom  F )
13 fores 5738 . . . 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 18978 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . 5  |-  ( ph  ->  K  e.  Top )
17 imassrn 5289 . . . . . 6  |-  ( F
" A )  C_  ran  F
18 frn 5674 . . . . . . 7  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
196, 18syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  U. K
)
2017, 19syl5ss 3476 . . . . 5  |-  ( ph  ->  ( F " A
)  C_  U. K )
214restuni 18899 . . . . 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 5727 . . . 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 19022 . . . 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 18671 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
2916, 28sylib 196 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
30 df-ima 4962 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
31 eqimss2 3518 . . . . 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 19023 . . . 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 1219 . . 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 2454 . . 3  |-  U. ( Kt  ( F " A ) )  =  U. ( Kt  ( F " A ) )
3736cnconn 19159 . 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 1219 1  |-  ( ph  ->  ( Kt  ( F " A ) )  e. 
Con )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758    C_ wss 3437   U.cuni 4200   dom cdm 4949   ran crn 4950    |` cres 4951   "cima 4952   Fun wfun 5521   -->wf 5523   -onto->wfo 5525   ` cfv 5527  (class class class)co 6201   ↾t crest 14479   Topctop 18631  TopOnctopon 18632    Cn ccn 18961   Conccon 19148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-fin 7425  df-fi 7773  df-rest 14481  df-topgen 14502  df-top 18636  df-bases 18638  df-topon 18639  df-cld 18756  df-cn 18964  df-con 19149
This theorem is referenced by:  tgpconcompeqg  19815  tgpconcomp  19816
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