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Theorem hmphindis 19345
Description: Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmphdis.1  |-  X  = 
U. J
Assertion
Ref Expression
hmphindis  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )

Proof of Theorem hmphindis
StepHypRef Expression
1 dfsn2 3885 . . 3  |-  { (/) }  =  { (/) ,  (/) }
2 indislem 18579 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
3 preq2 3950 . . . . . . . 8  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  (/) } )
43, 1syl6eqr 2488 . . . . . . 7  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
} )
52, 4syl5eqr 2484 . . . . . 6  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  A }  =  { (/)
} )
65breq2d 4299 . . . . 5  |-  ( (  _I  `  A )  =  (/)  ->  ( J  ~=  { (/) ,  A } 
<->  J  ~=  { (/) } ) )
76biimpac 486 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  ~=  {
(/) } )
8 hmph0 19343 . . . 4  |-  ( J  ~=  { (/) }  <->  J  =  { (/) } )
97, 8sylib 196 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) } )
109unieqd 4096 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  U. J  =  U. { (/) } )
11 hmphdis.1 . . . . 5  |-  X  = 
U. J
12 0ex 4417 . . . . . . 7  |-  (/)  e.  _V
1312unisn 4101 . . . . . 6  |-  U. { (/)
}  =  (/)
1413eqcomi 2442 . . . . 5  |-  (/)  =  U. { (/) }
1510, 11, 143eqtr4g 2495 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  X  =  (/) )
1615preq2d 3956 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  { (/) ,  X }  =  { (/)
,  (/) } )
171, 9, 163eqtr4a 2496 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) ,  X }
)
18 hmphen 19333 . . . . . 6  |-  ( J  ~=  { (/) ,  A }  ->  J  ~~  { (/)
,  A } )
1918adantr 465 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  {
(/) ,  A }
)
20 necom 2688 . . . . . . . 8  |-  ( (  _I  `  A )  =/=  (/)  <->  (/)  =/=  (  _I 
`  A ) )
21 fvex 5696 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
22 pr2nelem 8163 . . . . . . . . 9  |-  ( (
(/)  e.  _V  /\  (  _I  `  A )  e. 
_V  /\  (/)  =/=  (  _I  `  A ) )  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2312, 21, 22mp3an12 1304 . . . . . . . 8  |-  ( (/)  =/=  (  _I  `  A
)  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2420, 23sylbi 195 . . . . . . 7  |-  ( (  _I  `  A )  =/=  (/)  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2524adantl 466 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  (  _I  `  A
) }  ~~  2o )
262, 25syl5eqbrr 4321 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  A }  ~~  2o )
27 entr 7353 . . . . 5  |-  ( ( J  ~~  { (/) ,  A }  /\  { (/)
,  A }  ~~  2o )  ->  J  ~~  2o )
2819, 26, 27syl2anc 661 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  2o )
29 hmphtop1 19327 . . . . . . 7  |-  ( J  ~=  { (/) ,  A }  ->  J  e.  Top )
3029adantr 465 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e. 
Top )
3111toptopon 18513 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3230, 31sylib 196 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e.  (TopOn `  X )
)
33 en2top 18565 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
3432, 33syl 16 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  ( J 
~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
3528, 34mpbid 210 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
3635simpld 459 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  =  { (/) ,  X }
)
3717, 36pm2.61dane 2684 1  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967   (/)c0 3632   {csn 3872   {cpr 3874   U.cuni 4086   class class class wbr 4287    _I cid 4626   ` cfv 5413   2oc2o 6906    ~~ cen 7299   Topctop 18473  TopOnctopon 18474    ~= chmph 19302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-1o 6912  df-2o 6913  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-top 18478  df-topon 18481  df-cn 18806  df-hmeo 19303  df-hmph 19304
This theorem is referenced by: (None)
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