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Theorem strfvi 15741
 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 6165 . . . . 5 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
32eqcomd 2616 . . . 4 (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
43fveq2d 6107 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
5 strfvi.e . . . . 5 𝐸 = Slot 𝑁
65str0 15739 . . . 4 ∅ = (𝐸‘∅)
7 fvprc 6097 . . . 4 𝑆 ∈ V → (𝐸𝑆) = ∅)
8 fvprc 6097 . . . . 5 𝑆 ∈ V → ( I ‘𝑆) = ∅)
98fveq2d 6107 . . . 4 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
106, 7, 93eqtr4a 2670 . . 3 𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
114, 10pm2.61i 175 . 2 (𝐸𝑆) = (𝐸‘( I ‘𝑆))
121, 11eqtri 2632 1 𝑋 = (𝐸‘( I ‘𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   I cid 4948  ‘cfv 5804  Slot cslot 15694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699 This theorem is referenced by:  rlmscaf  19029  islidl  19032  lidlrsppropd  19051  rspsn  19075  ply1tmcl  19463  ply1scltm  19472  ply1sclf  19476  ply1scl0  19481  ply1scl1  19483  nrgtrg  22304
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