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Theorem indislem 17069
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 5786 . . 3  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
21preq2d 3892 . 2  |-  ( A  e.  _V  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A } )
3 dfsn2 3830 . . . 4  |-  { (/) }  =  { (/) ,  (/) }
43eqcomi 2442 . . 3  |-  { (/) ,  (/) }  =  { (/) }
5 fvprc 5725 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
65preq2d 3892 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  (/) } )
7 prprc2 3917 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  A }  =  { (/) } )
84, 6, 73eqtr4a 2496 . 2  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  A }
)
92, 8pm2.61i 159 1  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {csn 3816   {cpr 3817    _I cid 4496   ` cfv 5457
This theorem is referenced by:  indistop  17071  indisuni  17072  indiscld  17160  indiscon  17486  txindis  17671  hmphindis  17834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465
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