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Theorem indislem 16737
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 5579 . . 3  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
21preq2d 3713 . 2  |-  ( A  e.  _V  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A } )
3 dfsn2 3654 . . . 4  |-  { (/) }  =  { (/) ,  (/) }
43eqcomi 2287 . . 3  |-  { (/) ,  (/) }  =  { (/) }
5 fvprc 5519 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
65preq2d 3713 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  (/) } )
7 prprc2 3737 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  A }  =  { (/) } )
84, 6, 73eqtr4a 2341 . 2  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  A }
)
92, 8pm2.61i 156 1  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   {cpr 3641    _I cid 4304   ` cfv 5255
This theorem is referenced by:  indistop  16739  indisuni  16740  indiscld  16828  indiscon  17144  txindis  17328  hmphindis  17488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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