Theorem strfvi 15913

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 6255	. . . . 5 |- ( S e. _V -> ( _I ` S ) = S )
3	2	eqcomd 2628	. . . 4 |- ( S e. _V -> S = ( _I ` S ) )
4	3	fveq2d 6195	. . 3 |- ( S e. _V -> ( E ` S ) = ( E ` ( _I ` S ) ) )
5		strfvi.e	. . . . 5 |- E = Slot N
6	5	str0 15911	. . . 4 |- (/) = ( E ` (/) )
7		fvprc 6185	. . . 4 |- ( -. S e. _V -> ( E ` S ) = (/) )
8		fvprc 6185	. . . . 5 |- ( -. S e. _V -> ( _I ` S ) = (/) )
9	8	fveq2d 6195	. . . 4 |- ( -. S e. _V -> ( E ` ( _I ` S ) ) = ( E ` (/) ) )
10	6, 7, 9	3eqtr4a 2682	. . 3 |- ( -. S e. _V -> ( E ` S ) = ( E ` ( _I ` S ) ) )
11	4, 10	pm2.61i 176	. 2 |- ( E ` S ) = ( E ` ( _I ` S ) )
12	1, 11	eqtri 2644	1 |- X = ( E ` ( _I ` S ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   _I cid 5023   ` cfv 5888  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  rlmscaf  19208  islidl  19211  lidlrsppropd  19230  rspsn  19254  ply1tmcl  19642  ply1scltm  19651  ply1sclf  19655  ply1scl0  19660  ply1scl1  19662  nrgtrg  22494