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Theorem ids1 12723
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 5891 . . . . 5  |-  (  _I 
`  A )  e. 
_V
2 fvi 5938 . . . . 5  |-  ( (  _I  `  A )  e.  _V  ->  (  _I  `  (  _I  `  A ) )  =  (  _I  `  A
) )
31, 2ax-mp 5 . . . 4  |-  (  _I 
`  (  _I  `  A ) )  =  (  _I  `  A
)
43opeq2i 4194 . . 3  |-  <. 0 ,  (  _I  `  (  _I  `  A ) )
>.  =  <. 0 ,  (  _I  `  A
) >.
54sneqi 4013 . 2  |-  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }  =  { <. 0 ,  (  _I  `  A )
>. }
6 df-s1 12654 . 2  |-  <" (  _I  `  A ) ">  =  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }
7 df-s1 12654 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
85, 6, 73eqtr4ri 2469 1  |-  <" A ">  =  <" (  _I  `  A ) ">
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002   <.cop 4008    _I cid 4764   ` cfv 5601   0cc0 9538   <"cs1 12646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-s1 12654
This theorem is referenced by:  s1cli  12730  revs1  12855
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