Theorem strfvi 17160

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 6916	. . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
3	2	eqcomd 2742	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
4	3	fveq2d 6844	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
5		strfvi.e	. . . . 5 ⊢ 𝐸 = Slot 𝑁
6	5	str0 17159	. . . 4 ⊢ ∅ = (𝐸‘∅)
7		fvprc 6832	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
8		fvprc 6832	. . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅)
9	8	fveq2d 6844	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
10	6, 7, 9	3eqtr4a 2797	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
11	4, 10	pm2.61i 182	. 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))
12	1, 11	eqtri 2759	1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆))

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273   I cid 5525  ‘cfv 6498  Slot cslot 17151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-slot 17152

This theorem is referenced by:  rlmscaf  21202  islidl  21213  lidlrsppropd  21242  rspsn  21331  ply1tmcl  22237  ply1scltm  22246  ply1sclf  22250  nrgtrg  24655