strfvi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > strfvi		Structured version Visualization version GIF version

Theorem strfvi 16961

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 6883	. . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆)
3	2	eqcomd 2743	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆))
4	3	fveq2d 6815	. . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
5		strfvi.e	. . . . 5 ⊢ 𝐸 = Slot 𝑁
6	5	str0 16960	. . . 4 ⊢ ∅ = (𝐸‘∅)
7		fvprc 6803	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅)
8		fvprc 6803	. . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅)
9	8	fveq2d 6815	. . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅))
10	6, 7, 9	3eqtr4a 2803	. . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)))
11	4, 10	pm2.61i 182	. 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))
12	1, 11	eqtri 2765	1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∅c0 4267 I cid 5506 ‘cfv 6465 Slot cslot 16952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5238 ax-nul 5245 ax-pr 5367

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3405 df-v 3443 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4268 df-if 4472 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-br 5088 df-opab 5150 df-mpt 5171 df-id 5507 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-iota 6417 df-fun 6467 df-fv 6473 df-slot 16953

This theorem is referenced by: rlmscaf 20551 islidl 20554 lidlrsppropd 20573 rspsn 20597 ply1tmcl 21515 ply1scltm 21524 ply1sclf 21528 ply1scl0 21533 ply1scl1 21535 nrgtrg 23926

W3C validator