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Theorem ids1 11437
Description: Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
ids1  |-  <" A ">  =  <" (  _I  `  A ) ">

Proof of Theorem ids1
StepHypRef Expression
1 fvex 5539 . . . . 5  |-  (  _I 
`  A )  e. 
_V
2 fvi 5579 . . . . 5  |-  ( (  _I  `  A )  e.  _V  ->  (  _I  `  (  _I  `  A ) )  =  (  _I  `  A
) )
31, 2ax-mp 8 . . . 4  |-  (  _I 
`  (  _I  `  A ) )  =  (  _I  `  A
)
43opeq2i 3800 . . 3  |-  <. 0 ,  (  _I  `  (  _I  `  A ) )
>.  =  <. 0 ,  (  _I  `  A
) >.
54sneqi 3652 . 2  |-  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }  =  { <. 0 ,  (  _I  `  A )
>. }
6 df-s1 11411 . 2  |-  <" (  _I  `  A ) ">  =  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }
7 df-s1 11411 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
85, 6, 73eqtr4ri 2314 1  |-  <" A ">  =  <" (  _I  `  A ) ">
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    _I cid 4304   ` cfv 5255   0cc0 8737   <"cs1 11405
This theorem is referenced by:  s1cli  11443  revs1  11483
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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  df-s1 11411
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