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Theorem indislem 17019
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 5742 . . 3  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
21preq2d 3850 . 2  |-  ( A  e.  _V  ->  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A } )
3 dfsn2 3788 . . . 4  |-  { (/) }  =  { (/) ,  (/) }
43eqcomi 2408 . . 3  |-  { (/) ,  (/) }  =  { (/) }
5 fvprc 5681 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
65preq2d 3850 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  (/) } )
7 prprc2 3875 . . 3  |-  ( -.  A  e.  _V  ->  {
(/) ,  A }  =  { (/) } )
84, 6, 73eqtr4a 2462 . 2  |-  ( -.  A  e.  _V  ->  {
(/) ,  (  _I  `  A ) }  =  { (/) ,  A }
)
92, 8pm2.61i 158 1  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   {csn 3774   {cpr 3775    _I cid 4453   ` cfv 5413
This theorem is referenced by:  indistop  17021  indisuni  17022  indiscld  17110  indiscon  17434  txindis  17619  hmphindis  17782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421
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