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Theorem strfvi 16136
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 6419 . . . . 5 (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
32eqcomd 2767 . . . 4 (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
43fveq2d 6358 . . 3 (𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
5 strfvi.e . . . . 5 𝐸 = Slot 𝑁
65str0 16134 . . . 4 ∅ = (𝐸‘∅)
7 fvprc 6348 . . . 4 𝑆 ∈ V → (𝐸𝑆) = ∅)
8 fvprc 6348 . . . . 5 𝑆 ∈ V → ( I ‘𝑆) = ∅)
98fveq2d 6358 . . . 4 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
106, 7, 93eqtr4a 2821 . . 3 𝑆 ∈ V → (𝐸𝑆) = (𝐸‘( I ‘𝑆)))
114, 10pm2.61i 176 . 2 (𝐸𝑆) = (𝐸‘( I ‘𝑆))
121, 11eqtri 2783 1 𝑋 = (𝐸‘( I ‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2140  Vcvv 3341  c0 4059   I cid 5174  cfv 6050  Slot cslot 16079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-slot 16084
This theorem is referenced by:  rlmscaf  19431  islidl  19434  lidlrsppropd  19453  rspsn  19477  ply1tmcl  19865  ply1scltm  19874  ply1sclf  19878  ply1scl0  19883  ply1scl1  19885  nrgtrg  22716
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