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Theorem hmphindis 20026
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 4033 . . 3  |-  { (/) }  =  { (/) ,  (/) }
2 indislem 19260 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
3 preq2 4100 . . . . . . . 8  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  (/) } )
43, 1syl6eqr 2519 . . . . . . 7  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
} )
52, 4syl5eqr 2515 . . . . . 6  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  A }  =  { (/)
} )
65breq2d 4452 . . . . 5  |-  ( (  _I  `  A )  =  (/)  ->  ( J  ~=  { (/) ,  A } 
<->  J  ~=  { (/) } ) )
76biimpac 486 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  ~=  {
(/) } )
8 hmph0 20024 . . . 4  |-  ( J  ~=  { (/) }  <->  J  =  { (/) } )
97, 8sylib 196 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) } )
109unieqd 4248 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  U. J  =  U. { (/) } )
11 hmphdis.1 . . . . 5  |-  X  = 
U. J
12 0ex 4570 . . . . . . 7  |-  (/)  e.  _V
1312unisn 4253 . . . . . 6  |-  U. { (/)
}  =  (/)
1413eqcomi 2473 . . . . 5  |-  (/)  =  U. { (/) }
1510, 11, 143eqtr4g 2526 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  X  =  (/) )
1615preq2d 4106 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  { (/) ,  X }  =  { (/)
,  (/) } )
171, 9, 163eqtr4a 2527 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) ,  X }
)
18 hmphen 20014 . . . . . 6  |-  ( J  ~=  { (/) ,  A }  ->  J  ~~  { (/)
,  A } )
1918adantr 465 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  {
(/) ,  A }
)
20 necom 2729 . . . . . . . 8  |-  ( (  _I  `  A )  =/=  (/)  <->  (/)  =/=  (  _I 
`  A ) )
21 fvex 5867 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
22 pr2nelem 8371 . . . . . . . . 9  |-  ( (
(/)  e.  _V  /\  (  _I  `  A )  e. 
_V  /\  (/)  =/=  (  _I  `  A ) )  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2312, 21, 22mp3an12 1309 . . . . . . . 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 4474 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  A }  ~~  2o )
27 entr 7557 . . . . 5  |-  ( ( J  ~~  { (/) ,  A }  /\  { (/)
,  A }  ~~  2o )  ->  J  ~~  2o )
2819, 26, 27syl2anc 661 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  2o )
29 hmphtop1 20008 . . . . . . 7  |-  ( J  ~=  { (/) ,  A }  ->  J  e.  Top )
3029adantr 465 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e. 
Top )
3111toptopon 19194 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3230, 31sylib 196 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e.  (TopOn `  X )
)
33 en2top 19246 . . . . 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 2778 1  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106   (/)c0 3778   {csn 4020   {cpr 4022   U.cuni 4238   class class class wbr 4440    _I cid 4783   ` cfv 5579   2oc2o 7114    ~~ cen 7503   Topctop 19154  TopOnctopon 19155    ~= chmph 19983
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-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-1o 7120  df-2o 7121  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-top 19159  df-topon 19162  df-cn 19487  df-hmeo 19984  df-hmph 19985
This theorem is referenced by: (None)
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