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Theorem strfvi 15834
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 6212 . . . . 5 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
32eqcomd 2627 . . . 4 (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
43fveq2d 6152 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
5 strfvi.e . . . . 5 𝐸 = Slot 𝑁
65str0 15832 . . . 4 ∅ = (𝐸‘∅)
7 fvprc 6142 . . . 4 𝑆 ∈ V → (𝐸𝑆) = ∅)
8 fvprc 6142 . . . . 5 𝑆 ∈ V → ( I ‘𝑆) = ∅)
98fveq2d 6152 . . . 4 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
106, 7, 93eqtr4a 2681 . . 3 𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
114, 10pm2.61i 176 . 2 (𝐸𝑆) = (𝐸‘( I ‘𝑆))
121, 11eqtri 2643 1 𝑋 = (𝐸‘( I ‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891   I cid 4984  cfv 5847  Slot cslot 15780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-slot 15785
This theorem is referenced by:  rlmscaf  19127  islidl  19130  lidlrsppropd  19149  rspsn  19173  ply1tmcl  19561  ply1scltm  19570  ply1sclf  19574  ply1scl0  19579  ply1scl1  19581  nrgtrg  22404
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