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Theorem hmphindis 19495
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 3991 . . 3  |-  { (/) }  =  { (/) ,  (/) }
2 indislem 18729 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
3 preq2 4056 . . . . . . . 8  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  (/) } )
43, 1syl6eqr 2510 . . . . . . 7  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
} )
52, 4syl5eqr 2506 . . . . . 6  |-  ( (  _I  `  A )  =  (/)  ->  { (/) ,  A }  =  { (/)
} )
65breq2d 4405 . . . . 5  |-  ( (  _I  `  A )  =  (/)  ->  ( J  ~=  { (/) ,  A } 
<->  J  ~=  { (/) } ) )
76biimpac 486 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  ~=  {
(/) } )
8 hmph0 19493 . . . 4  |-  ( J  ~=  { (/) }  <->  J  =  { (/) } )
97, 8sylib 196 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) } )
109unieqd 4202 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  U. J  =  U. { (/) } )
11 hmphdis.1 . . . . 5  |-  X  = 
U. J
12 0ex 4523 . . . . . . 7  |-  (/)  e.  _V
1312unisn 4207 . . . . . 6  |-  U. { (/)
}  =  (/)
1413eqcomi 2464 . . . . 5  |-  (/)  =  U. { (/) }
1510, 11, 143eqtr4g 2517 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  X  =  (/) )
1615preq2d 4062 . . 3  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  { (/) ,  X }  =  { (/)
,  (/) } )
171, 9, 163eqtr4a 2518 . 2  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =  (/) )  ->  J  =  { (/) ,  X }
)
18 hmphen 19483 . . . . . 6  |-  ( J  ~=  { (/) ,  A }  ->  J  ~~  { (/)
,  A } )
1918adantr 465 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  {
(/) ,  A }
)
20 necom 2717 . . . . . . . 8  |-  ( (  _I  `  A )  =/=  (/)  <->  (/)  =/=  (  _I 
`  A ) )
21 fvex 5802 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
22 pr2nelem 8275 . . . . . . . . 9  |-  ( (
(/)  e.  _V  /\  (  _I  `  A )  e. 
_V  /\  (/)  =/=  (  _I  `  A ) )  ->  { (/) ,  (  _I  `  A ) }  ~~  2o )
2312, 21, 22mp3an12 1305 . . . . . . . 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 4427 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  { (/) ,  A }  ~~  2o )
27 entr 7464 . . . . 5  |-  ( ( J  ~~  { (/) ,  A }  /\  { (/)
,  A }  ~~  2o )  ->  J  ~~  2o )
2819, 26, 27syl2anc 661 . . . 4  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  ~~  2o )
29 hmphtop1 19477 . . . . . . 7  |-  ( J  ~=  { (/) ,  A }  ->  J  e.  Top )
3029adantr 465 . . . . . 6  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e. 
Top )
3111toptopon 18663 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3230, 31sylib 196 . . . . 5  |-  ( ( J  ~=  { (/) ,  A }  /\  (  _I  `  A )  =/=  (/) )  ->  J  e.  (TopOn `  X )
)
33 en2top 18715 . . . . 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 2766 1  |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
,  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071   (/)c0 3738   {csn 3978   {cpr 3980   U.cuni 4192   class class class wbr 4393    _I cid 4732   ` cfv 5519   2oc2o 7017    ~~ cen 7410   Topctop 18623  TopOnctopon 18624    ~= chmph 19452
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-1o 7023  df-2o 7024  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-top 18628  df-topon 18631  df-cn 18956  df-hmeo 19453  df-hmph 19454
This theorem is referenced by: (None)
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