MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmphindis Unicode version

Theorem hmphindis 17588
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 3730 . . 3  |-  { (/) }  =  { (/) ,  (/) }
2 indislem 16837 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
3 preq2 3783 . . . . . . . 8  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  (/) } )
43, 1syl6eqr 2408 . . . . . . 7  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
} )
52, 4syl5eqr 2404 . . . . . 6  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  A }  =  { (/)
} )
65breq2d 4114 . . . . 5  |-  ( (  _I  `  A )  =  (/)  ->  ( J  ~=  { (/) ,  A } 
<->  J  ~=  { (/) } ) )
76biimpac 472 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  ~=  {
(/) } )
8 hmph0 17586 . . . 4  |-  ( J  ~=  { (/) }  <->  J  =  { (/) } )
97, 8sylib 188 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) } )
109unieqd 3917 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  U. J  =  U. { (/) } )
11 hmphdis.1 . . . . 5  |-  X  = 
U. J
12 0ex 4229 . . . . . . 7  |-  (/)  e.  _V
1312unisn 3922 . . . . . 6  |-  U. { (/)
}  =  (/)
1413eqcomi 2362 . . . . 5  |-  (/)  =  U. { (/) }
1510, 11, 143eqtr4g 2415 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  X  =  (/) )
1615preq2d 3789 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  { (/) ,  X }  =  { (/)
,  (/) } )
171, 9, 163eqtr4a 2416 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) ,  X }
)
18 hmphen 17576 . . . . . 6  |-  ( J  ~=  { (/) ,  A }  ->  J  ~~  { (/)
,  A } )
1918adantr 451 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  {
(/) ,  A }
)
20 necom 2602 . . . . . . . 8  |-  ( (  _I  `  A )  =/=  (/)  <->  (/)  =/=  (  _I 
`  A ) )
21 fvex 5619 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
22 pr2nelem 7721 . . . . . . . . 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 4136 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  A }  ~~  2o )
27 entr 6998 . . . . 5  |-  ( ( J  ~~  { (/) ,  A }  /\  { (/)
,  A }  ~~  2o )  ->  J  ~~  2o )
2819, 26, 27syl2anc 642 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  2o )
29 hmphtop1 17570 . . . . . . 7  |-  ( J  ~=  { (/) ,  A }  ->  J  e.  Top )
3029adantr 451 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e. 
Top )
3111toptopon 16771 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3230, 31sylib 188 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e.  (TopOn `  X )
)
33 en2top 16823 . . . . 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 2599 1  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   (/)c0 3531   {csn 3716   {cpr 3717   U.cuni 3906   class class class wbr 4102    _I cid 4383   ` cfv 5334   2oc2o 6557    ~~ cen 6945   Topctop 16731  TopOnctopon 16732    ~= chmph 17545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-1o 6563  df-2o 6564  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-card 7659  df-top 16736  df-topon 16739  df-cn 17057  df-hmeo 17546  df-hmph 17547
  Copyright terms: Public domain W3C validator