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Theorem strfvi 13499
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 5775 . . . . 5  |-  ( S  e.  _V  ->  (  _I  `  S )  =  S )
32eqcomd 2440 . . . 4  |-  ( S  e.  _V  ->  S  =  (  _I  `  S
) )
43fveq2d 5724 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
5 strfvi.e . . . . 5  |-  E  = Slot 
N
65str0 13497 . . . 4  |-  (/)  =  ( E `  (/) )
7 fvprc 5714 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
8 fvprc 5714 . . . . 5  |-  ( -.  S  e.  _V  ->  (  _I  `  S )  =  (/) )
98fveq2d 5724 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  (  _I 
`  S ) )  =  ( E `  (/) ) )
106, 7, 93eqtr4a 2493 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
114, 10pm2.61i 158 . 2  |-  ( E `
 S )  =  ( E `  (  _I  `  S ) )
121, 11eqtri 2455 1  |-  X  =  ( E `  (  _I  `  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620    _I cid 4485   ` cfv 5446  Slot cslot 13460
This theorem is referenced by:  rlmscaf  16271  islidl  16274  lidlrsppropd  16293  rspsn  16317  ply1tmcl  16656  ply1scltm  16665  ply1sclf  16669  ply1scl0  16673  ply1scl1  16675  nrgtrg  18717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-slot 13465
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