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Theorem strfvi 14318
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 5849 . . . . 5  |-  ( S  e.  _V  ->  (  _I  `  S )  =  S )
32eqcomd 2459 . . . 4  |-  ( S  e.  _V  ->  S  =  (  _I  `  S
) )
43fveq2d 5795 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
5 strfvi.e . . . . 5  |-  E  = Slot 
N
65str0 14316 . . . 4  |-  (/)  =  ( E `  (/) )
7 fvprc 5785 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
8 fvprc 5785 . . . . 5  |-  ( -.  S  e.  _V  ->  (  _I  `  S )  =  (/) )
98fveq2d 5795 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  (  _I 
`  S ) )  =  ( E `  (/) ) )
106, 7, 93eqtr4a 2518 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
114, 10pm2.61i 164 . 2  |-  ( E `
 S )  =  ( E `  (  _I  `  S ) )
121, 11eqtri 2480 1  |-  X  =  ( E `  (  _I  `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737    _I cid 4731   ` cfv 5518  Slot cslot 14277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-slot 14282
This theorem is referenced by:  rlmscaf  17397  islidl  17401  lidlrsppropd  17420  rspsn  17444  ply1tmcl  17835  ply1scltm  17844  ply1sclf  17848  ply1scl0  17853  ply1scl1  17855  nrgtrg  20388
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