Theorem strfvi 17400

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 6987	. . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
3	2	eqcomd 2769	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
4	3	fveq2d 6915	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
5		strfvi.e	. . . . 5 ⊢ 𝐸 = Slot 𝑁
6	5	str0 17399	. . . 4 ⊢ ∅ = (𝐸‘∅)
7		fvprc 6903	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
8		fvprc 6903	. . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅)
9	8	fveq2d 6915	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
10	6, 7, 9	3eqtr4a 2824	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
11	4, 10	pm2.61i 184	. 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))
12	1, 11	eqtri 2786	1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281   I cid 5550  ‘cfv 6561  Slot cslot 17391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-slot 17392

This theorem is used by:  rlmscaf  21515  islidl  21527  lidlrsppropd  21565  rspsn  21690  ply1tmcl  22624  ply1scltm  22633  ply1sclf  22637  nrgtrg  25042