Theorem setsnid 15158

Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
setsid.e	|- E = Slot ( E ` ndx )
setsnid.n	|- ( E ` ndx ) =/= D

Assertion

Ref	Expression
setsnid	|- ( E ` W ) = ( E ` ( W sSet <. D , C >. ) )

Proof of Theorem setsnid

Step	Hyp	Ref	Expression
1		setsid.e	. . . 4 |- E = Slot ( E ` ndx )
2		id 23	. . . 4 |- ( W e. _V -> W e. _V )
3	1, 2	strfvnd 15129	. . 3 |- ( W e. _V -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
4		ovex 6331	. . . . 5 |- ( W sSet <. D , C >. ) e. _V
5	4, 1	strfvn 15131	. . . 4 |- ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
6		setsres 15144	. . . . . 6 |- ( W e. _V -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
7	6	fveq1d 5881	. . . . 5 |- ( W e. _V -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
8		fvex 5889	. . . . . . 7 |- ( E ` ndx ) e. _V
9		setsnid.n	. . . . . . 7 |- ( E ` ndx ) =/= D
10		eldifsn 4123	. . . . . . 7 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
11	8, 9, 10	mpbir2an 929	. . . . . 6 |- ( E ` ndx ) e. ( _V \ { D } )
12		fvres 5893	. . . . . 6 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) )
13	11, 12	ax-mp 5	. . . . 5 |- ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
14		fvres 5893	. . . . . 6 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
15	11, 14	ax-mp 5	. . . . 5 |- ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) )
16	7, 13, 15	3eqtr3g 2487	. . . 4 |- ( W e. _V -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
17	5, 16	syl5eq 2476	. . 3 |- ( W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( W ` ( E ` ndx ) ) )
18	3, 17	eqtr4d 2467	. 2 |- ( W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
19	1	str0 15154	. . 3 |- (/) = ( E ` (/) )
20		fvprc 5873	. . 3 |- ( -. W e. _V -> ( E ` W ) = (/) )
21		reldmsets 15137	. . . . 5 |- Rel dom sSet
22	21	ovprc1 6334	. . . 4 |- ( -. W e. _V -> ( W sSet <. D , C >. ) = (/) )
23	22	fveq2d 5883	. . 3 |- ( -. W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( E ` (/) ) )
24	19, 20, 23	3eqtr4a 2490	. 2 |- ( -. W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
25	18, 24	pm2.61i 168	1 |- ( E ` W ) = ( E ` ( W sSet <. D , C >. ) )

Syntax hints:   -. wn 3   = wceq 1438   e. wcel 1869   =/= wne 2619   _Vcvv 3082   \ cdif 3434   (/)c0 3762   {csn 3997   <.cop 4003   |` cres 4853   ` cfv 5599  (class class class)co 6303   ndxcnx 15111   sSet csts 15112  Slot cslot 15113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-res 4863  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-slot 15118  df-sets 15120

This theorem is referenced by:  resslem  15175  oppchomfval  15612  oppcbas  15616  rescbas  15727  rescco  15730  rescabs  15731  odubas  16372  oppglem  16994  mgplem  17721  opprlem  17849  sralem  18393  srasca  18397  sravsca  18398  opsrbaslem  18694  zlmlem  19080  zlmsca  19084  znbaslem  19101  thlbas  19251  thlle  19252  matbas  19430  matplusg  19431  matsca  19432  matvsca  19433  tuslem  21274  setsmsbas  21482  setsmsds  21483  tnglem  21640  tngds  21648  ttgval  24897  ttglem  24898  cchhllem  24909  resvlem  28596  zlmds  28770  zlmtset  28771  hlhilslem  35472  uhgrepe  39032  cznrnglem  39297  cznabel  39298  cznrng  39299