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Theorem ids1 12569
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 5874 . . . . 5  |-  (  _I 
`  A )  e. 
_V
2 fvi 5922 . . . . 5  |-  ( (  _I  `  A )  e.  _V  ->  (  _I  `  (  _I  `  A ) )  =  (  _I  `  A
) )
31, 2ax-mp 5 . . . 4  |-  (  _I 
`  (  _I  `  A ) )  =  (  _I  `  A
)
43opeq2i 4217 . . 3  |-  <. 0 ,  (  _I  `  (  _I  `  A ) )
>.  =  <. 0 ,  (  _I  `  A
) >.
54sneqi 4038 . 2  |-  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }  =  { <. 0 ,  (  _I  `  A )
>. }
6 df-s1 12507 . 2  |-  <" (  _I  `  A ) ">  =  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }
7 df-s1 12507 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
85, 6, 73eqtr4ri 2507 1  |-  <" A ">  =  <" (  _I  `  A ) ">
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   <.cop 4033    _I cid 4790   ` cfv 5586   0cc0 9488   <"cs1 12499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-s1 12507
This theorem is referenced by:  s1cli  12575  revs1  12698
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