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Theorem strfvi 16528
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 = (𝐸𝑆)
Assertion
Ref Expression
strfvi 𝑋 = (𝐸‘( I ‘𝑆))

Proof of Theorem strfvi
StepHypRef Expression
1 strfvi.x . 2 𝑋 = (𝐸𝑆)
2 fvi 6722 . . . . 5 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
32eqcomd 2828 . . . 4 (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
43fveq2d 6656 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
5 strfvi.e . . . . 5 𝐸 = Slot 𝑁
65str0 16526 . . . 4 ∅ = (𝐸‘∅)
7 fvprc 6645 . . . 4 𝑆 ∈ V → (𝐸𝑆) = ∅)
8 fvprc 6645 . . . . 5 𝑆 ∈ V → ( I ‘𝑆) = ∅)
98fveq2d 6656 . . . 4 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
106, 7, 93eqtr4a 2883 . . 3 𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
114, 10pm2.61i 185 . 2 (𝐸𝑆) = (𝐸‘( I ‘𝑆))
121, 11eqtri 2845 1 𝑋 = (𝐸‘( I ‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265   I cid 5436  cfv 6334  Slot cslot 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-slot 16478
This theorem is referenced by:  rlmscaf  19972  islidl  19975  lidlrsppropd  19994  rspsn  20018  ply1tmcl  20899  ply1scltm  20908  ply1sclf  20912  ply1scl0  20917  ply1scl1  20919  nrgtrg  23294
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