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Theorem hmphindis 17488
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 3654 . . 3  |-  { (/) }  =  { (/) ,  (/) }
2 indislem 16737 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
3 preq2 3707 . . . . . . . 8  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  (/) } )
43, 1syl6eqr 2333 . . . . . . 7  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
} )
52, 4syl5eqr 2329 . . . . . 6  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  A }  =  { (/)
} )
65breq2d 4035 . . . . 5  |-  ( (  _I  `  A )  =  (/)  ->  ( J  ~=  { (/) ,  A } 
<->  J  ~=  { (/) } ) )
76biimpac 472 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  ~=  {
(/) } )
8 hmph0 17486 . . . 4  |-  ( J  ~=  { (/) }  <->  J  =  { (/) } )
97, 8sylib 188 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) } )
109unieqd 3838 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  U. J  =  U. { (/) } )
11 hmphdis.1 . . . . 5  |-  X  = 
U. J
12 0ex 4150 . . . . . . 7  |-  (/)  e.  _V
1312unisn 3843 . . . . . 6  |-  U. { (/)
}  =  (/)
1413eqcomi 2287 . . . . 5  |-  (/)  =  U. { (/) }
1510, 11, 143eqtr4g 2340 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  X  =  (/) )
1615preq2d 3713 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  { (/) ,  X }  =  { (/)
,  (/) } )
171, 9, 163eqtr4a 2341 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) ,  X }
)
18 hmphen 17476 . . . . . 6  |-  ( J  ~=  { (/) ,  A }  ->  J  ~~  { (/)
,  A } )
1918adantr 451 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  {
(/) ,  A }
)
20 necom 2527 . . . . . . . 8  |-  ( (  _I  `  A )  =/=  (/)  <->  (/)  =/=  (  _I 
`  A ) )
21 fvex 5539 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
22 pr2nelem 7634 . . . . . . . . 9  |-  ( (
(/)  e.  _V  /\  (  _I  `  A )  e. 
_V  /\  (/)  =/=  (  _I  `  A ) )  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2312, 21, 22mp3an12 1267 . . . . . . . 8  |-  ( (/)  =/=  (  _I  `  A
)  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2420, 23sylbi 187 . . . . . . 7  |-  ( (  _I  `  A )  =/=  (/)  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2524adantl 452 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  (  _I  `  A
) }  ~~  2o )
262, 25syl5eqbrr 4057 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  A }  ~~  2o )
27 entr 6913 . . . . 5  |-  ( ( J  ~~  { (/) ,  A }  /\  { (/)
,  A }  ~~  2o )  ->  J  ~~  2o )
2819, 26, 27syl2anc 642 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  2o )
29 hmphtop1 17470 . . . . . . 7  |-  ( J  ~=  { (/) ,  A }  ->  J  e.  Top )
3029adantr 451 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e. 
Top )
3111toptopon 16671 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3230, 31sylib 188 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e.  (TopOn `  X )
)
33 en2top 16723 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
3432, 33syl 15 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  ( J 
~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
3528, 34mpbid 201 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
3635simpld 445 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  =  { (/) ,  X }
)
3717, 36pm2.61dane 2524 1  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640   {cpr 3641   U.cuni 3827   class class class wbr 4023    _I cid 4304   ` cfv 5255   2oc2o 6473    ~~ cen 6860   Topctop 16631  TopOnctopon 16632    ~= chmph 17445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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