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Theorem indislem 19260
Description: A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indislem  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }

Proof of Theorem indislem
StepHypRef Expression
1 fvi 5915 . . 3  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
21preq2d 4106 . 2  |-  ( A  e.  _V  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A } )
3 dfsn2 4033 . . . 4  |-  { (/) }  =  { (/) ,  (/) }
43eqcomi 2473 . . 3  |-  { (/) ,  (/) }  =  { (/) }
5 fvprc 5851 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
65preq2d 4106 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  (/) } )
7 prprc2 4131 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  A }  =  { (/) } )
84, 6, 73eqtr4a 2527 . 2  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  A }
)
92, 8pm2.61i 164 1  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762   _Vcvv 3106   (/)c0 3778   {csn 4020   {cpr 4022    _I cid 4783   ` cfv 5579
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587
This theorem is referenced by:  indistop  19262  indisuni  19263  indiscld  19351  indiscon  19678  txindis  19863  hmphindis  20026
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