strfvi - Metamath Proof Explorer

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Theorem strfvi 17167

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 6940	. . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
3	2	eqcomd 2736	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
4	3	fveq2d 6865	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
5		strfvi.e	. . . . 5 ⊢ 𝐸 = Slot 𝑁
6	5	str0 17166	. . . 4 ⊢ ∅ = (𝐸‘∅)
7		fvprc 6853	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
8		fvprc 6853	. . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅)
9	8	fveq2d 6865	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
10	6, 7, 9	3eqtr4a 2791	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
11	4, 10	pm2.61i 182	. 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))
12	1, 11	eqtri 2753	1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ∅c0 4299 I cid 5535 ‘cfv 6514 Slot cslot 17158

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-slot 17159

This theorem is referenced by: rlmscaf 21121 islidl 21132 lidlrsppropd 21161 rspsn 21250 ply1tmcl 22165 ply1scltm 22174 ply1sclf 22178 ply1scl0OLD 22184 ply1scl1OLD 22187 nrgtrg 24585

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