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Theorem strfvi 13186
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 5579 . . . . 5  |-  ( S  e.  _V  ->  (  _I  `  S )  =  S )
32eqcomd 2288 . . . 4  |-  ( S  e.  _V  ->  S  =  (  _I  `  S
) )
43fveq2d 5529 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
5 strfvi.e . . . . 5  |-  E  = Slot 
N
65str0 13184 . . . 4  |-  (/)  =  ( E `  (/) )
7 fvprc 5519 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
8 fvprc 5519 . . . . 5  |-  ( -.  S  e.  _V  ->  (  _I  `  S )  =  (/) )
98fveq2d 5529 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  (  _I 
`  S ) )  =  ( E `  (/) ) )
106, 7, 93eqtr4a 2341 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
114, 10pm2.61i 156 . 2  |-  ( E `
 S )  =  ( E `  (  _I  `  S ) )
121, 11eqtri 2303 1  |-  X  =  ( E `  (  _I  `  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    _I cid 4304   ` cfv 5255  Slot cslot 13147
This theorem is referenced by:  rlmscaf  15960  islidl  15963  lidlrsppropd  15982  rspsn  16006  ply1tmcl  16348  ply1scltm  16357  ply1sclf  16361  ply1scl0  16365  ply1scl1  16367  nrgtrg  18200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152
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