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Theorem ids1 12789
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 5889 . . . . 5  |-  (  _I 
`  A )  e. 
_V
2 fvi 5937 . . . . 5  |-  ( (  _I  `  A )  e.  _V  ->  (  _I  `  (  _I  `  A ) )  =  (  _I  `  A
) )
31, 2ax-mp 5 . . . 4  |-  (  _I 
`  (  _I  `  A ) )  =  (  _I  `  A
)
43opeq2i 4162 . . 3  |-  <. 0 ,  (  _I  `  (  _I  `  A ) )
>.  =  <. 0 ,  (  _I  `  A
) >.
54sneqi 3970 . 2  |-  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }  =  { <. 0 ,  (  _I  `  A )
>. }
6 df-s1 12714 . 2  |-  <" (  _I  `  A ) ">  =  { <. 0 ,  (  _I  `  (  _I  `  A
) ) >. }
7 df-s1 12714 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
85, 6, 73eqtr4ri 2504 1  |-  <" A ">  =  <" (  _I  `  A ) ">
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   _Vcvv 3031   {csn 3959   <.cop 3965    _I cid 4749   ` cfv 5589   0cc0 9557   <"cs1 12706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-s1 12714
This theorem is referenced by:  s1cli  12796  revs1  12924
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