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Theorem strfvi 13434
Description: Structure slot extractors cannot distinguish between proper classes and  (/), so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
strfvi.e  |-  E  = Slot 
N
strfvi.x  |-  X  =  ( E `  S
)
Assertion
Ref Expression
strfvi  |-  X  =  ( E `  (  _I  `  S ) )

Proof of Theorem strfvi
StepHypRef Expression
1 strfvi.x . 2  |-  X  =  ( E `  S
)
2 fvi 5722 . . . . 5  |-  ( S  e.  _V  ->  (  _I  `  S )  =  S )
32eqcomd 2392 . . . 4  |-  ( S  e.  _V  ->  S  =  (  _I  `  S
) )
43fveq2d 5672 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
5 strfvi.e . . . . 5  |-  E  = Slot 
N
65str0 13432 . . . 4  |-  (/)  =  ( E `  (/) )
7 fvprc 5662 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
8 fvprc 5662 . . . . 5  |-  ( -.  S  e.  _V  ->  (  _I  `  S )  =  (/) )
98fveq2d 5672 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  (  _I 
`  S ) )  =  ( E `  (/) ) )
106, 7, 93eqtr4a 2445 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
114, 10pm2.61i 158 . 2  |-  ( E `
 S )  =  ( E `  (  _I  `  S ) )
121, 11eqtri 2407 1  |-  X  =  ( E `  (  _I  `  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2899   (/)c0 3571    _I cid 4434   ` cfv 5394  Slot cslot 13395
This theorem is referenced by:  rlmscaf  16206  islidl  16209  lidlrsppropd  16228  rspsn  16252  ply1tmcl  16591  ply1scltm  16600  ply1sclf  16604  ply1scl0  16608  ply1scl1  16610  nrgtrg  18596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-slot 13400
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