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Theorem ids1 11753
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 5744 . . . . 5  |-  (  _I 
`  A )  e. 
_V
2 fvi 5785 . . . . 5  |-  ( (  _I  `  A )  e.  _V  ->  (  _I  `  (  _I  `  A ) )  =  (  _I  `  A
) )
31, 2ax-mp 8 . . . 4  |-  (  _I 
`  (  _I  `  A ) )  =  (  _I  `  A
)
43opeq2i 3990 . . 3  |-  <. 0 ,  (  _I  `  (  _I  `  A ) )
>.  =  <. 0 ,  (  _I  `  A
) >.
54sneqi 3828 . 2  |-  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }  =  { <. 0 ,  (  _I  `  A )
>. }
6 df-s1 11727 . 2  |-  <" (  _I  `  A ) ">  =  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }
7 df-s1 11727 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
85, 6, 73eqtr4ri 2469 1  |-  <" A ">  =  <" (  _I  `  A ) ">
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   <.cop 3819    _I cid 4495   ` cfv 5456   0cc0 8992   <"cs1 11721
This theorem is referenced by:  s1cli  11759  revs1  11799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-s1 11727
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