Theorem strfvi 17246

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

Step	Hyp	Ref	Expression
1		strfvi.x	. 2 ⊢ 𝑋 = (𝐸‘𝑆)
2		fvi 6955	. . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
3	2	eqcomd 2775	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
4	3	fveq2d 6883	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
5		strfvi.e	. . . . 5 ⊢ 𝐸 = Slot 𝑁
6	5	str0 17245	. . . 4 ⊢ ∅ = (𝐸‘∅)
7		fvprc 6871	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
8		fvprc 6871	. . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅)
9	8	fveq2d 6883	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
10	6, 7, 9	3eqtr4a 2830	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
11	4, 10	pm2.61i 184	. 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))
12	1, 11	eqtri 2792	1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆))

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294   I cid 5553  ‘cfv 6533  Slot cslot 17237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541  df-slot 17238

This theorem is referenced by:  rlmscaf  21302  islidl  21314  lidlrsppropd  21348  rspsn  21466  ply1tmcl  22398  ply1scltm  22407  ply1sclf  22411  nrgtrg  24812