Theorem setsnid 14216

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 22	. . . 4 |- ( W e. _V -> W e. _V )
3	1, 2	strfvnd 14189	. . 3 |- ( W e. _V -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
4		ovex 6116	. . . . 5 |- ( W sSet <. D , C >. ) e. _V
5	4, 1	strfvn 14191	. . . 4 |- ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
6		setsres 14202	. . . . . 6 |- ( W e. _V -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
7	6	fveq1d 5693	. . . . 5 |- ( W e. _V -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
8		fvex 5701	. . . . . . 7 |- ( E ` ndx ) e. _V
9		setsnid.n	. . . . . . 7 |- ( E ` ndx ) =/= D
10		eldifsn 4000	. . . . . . 7 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
11	8, 9, 10	mpbir2an 911	. . . . . 6 |- ( E ` ndx ) e. ( _V \ { D } )
12		fvres 5704	. . . . . 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 5704	. . . . . 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 2498	. . . 4 |- ( W e. _V -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
17	5, 16	syl5eq 2487	. . 3 |- ( W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( W ` ( E ` ndx ) ) )
18	3, 17	eqtr4d 2478	. 2 |- ( W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
19	1	str0 14212	. . 3 |- (/) = ( E ` (/) )
20		fvprc 5685	. . 3 |- ( -. W e. _V -> ( E ` W ) = (/) )
21		reldmsets 14196	. . . . 5 |- Rel dom sSet
22	21	ovprc1 6119	. . . 4 |- ( -. W e. _V -> ( W sSet <. D , C >. ) = (/) )
23	22	fveq2d 5695	. . 3 |- ( -. W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( E ` (/) ) )
24	19, 20, 23	3eqtr4a 2501	. 2 |- ( -. W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
25	18, 24	pm2.61i 164	1 |- ( E ` W ) = ( E ` ( W sSet <. D , C >. ) )

Syntax hints:   -. wn 3   = wceq 1369   e. wcel 1756   =/= wne 2606   _Vcvv 2972   \ cdif 3325   (/)c0 3637   {csn 3877   <.cop 3883   |` cres 4842   ` cfv 5418  (class class class)co 6091   ndxcnx 14171   sSet csts 14172  Slot cslot 14173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-slot 14178  df-sets 14180

This theorem is referenced by:  resslem  14231  oppchomfval  14653  oppcbas  14657  rescbas  14742  rescco  14745  rescabs  14746  odubas  15303  oppglem  15865  mgplem  16596  opprlem  16720  sralem  17258  srasca  17262  sravsca  17263  opsrbaslem  17559  zlmlem  17948  zlmsca  17952  znbaslem  17971  thlbas  18121  thlle  18122  matbas  18314  matplusg  18315  matsca  18316  matvsca  18317  tuslem  19842  setsmsbas  20050  setsmsds  20051  tnglem  20226  tngds  20234  ttgval  23121  ttglem  23122  cchhllem  23133  resvlem  26299  zlmds  26393  zlmtset  26394  hlhilslem  35586