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Theorem indislem 19479
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 4101 . 2  |-  ( A  e.  _V  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A } )
3 dfsn2 4027 . . . 4  |-  { (/) }  =  { (/) ,  (/) }
43eqcomi 2456 . . 3  |-  { (/) ,  (/) }  =  { (/) }
5 fvprc 5850 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
65preq2d 4101 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  (/) } )
7 prprc2 4126 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  A }  =  { (/) } )
84, 6, 73eqtr4a 2510 . 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 1383    e. wcel 1804   _Vcvv 3095   (/)c0 3770   {csn 4014   {cpr 4016    _I cid 4780   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586
This theorem is referenced by:  indistop  19481  indisuni  19482  indiscld  19570  indiscon  19897  txindis  20113  hmphindis  20276
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