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Theorem strfvi 14676
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 5831 . . . . 5  |-  ( S  e.  _V  ->  (  _I  `  S )  =  S )
32eqcomd 2390 . . . 4  |-  ( S  e.  _V  ->  S  =  (  _I  `  S
) )
43fveq2d 5778 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
5 strfvi.e . . . . 5  |-  E  = Slot 
N
65str0 14674 . . . 4  |-  (/)  =  ( E `  (/) )
7 fvprc 5768 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
8 fvprc 5768 . . . . 5  |-  ( -.  S  e.  _V  ->  (  _I  `  S )  =  (/) )
98fveq2d 5778 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  (  _I 
`  S ) )  =  ( E `  (/) ) )
106, 7, 93eqtr4a 2449 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  ( E `  (  _I  `  S ) ) )
114, 10pm2.61i 164 . 2  |-  ( E `
 S )  =  ( E `  (  _I  `  S ) )
121, 11eqtri 2411 1  |-  X  =  ( E `  (  _I  `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1399    e. wcel 1826   _Vcvv 3034   (/)c0 3711    _I cid 4704   ` cfv 5496  Slot cslot 14633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-slot 14638
This theorem is referenced by:  rlmscaf  17967  islidl  17971  lidlrsppropd  17991  rspsn  18015  ply1tmcl  18426  ply1scltm  18435  ply1sclf  18439  ply1scl0  18444  ply1scl1  18446  nrgtrg  21283
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