Theorem setsnid 14212

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 14185	. . 3 |- ( W e. _V -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
4		ovex 6115	. . . . 5 |- ( W sSet <. D , C >. ) e. _V
5	4, 1	strfvn 14187	. . . 4 |- ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
6		setsres 14198	. . . . . 6 |- ( W e. _V -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
7	6	fveq1d 5690	. . . . 5 |- ( W e. _V -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
8		fvex 5698	. . . . . . 7 |- ( E ` ndx ) e. _V
9		setsnid.n	. . . . . . 7 |- ( E ` ndx ) =/= D
10		eldifsn 3997	. . . . . . 7 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
11	8, 9, 10	mpbir2an 906	. . . . . 6 |- ( E ` ndx ) e. ( _V \ { D } )
12		fvres 5701	. . . . . 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 5701	. . . . . 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 2496	. . . 4 |- ( W e. _V -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
17	5, 16	syl5eq 2485	. . 3 |- ( W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( W ` ( E ` ndx ) ) )
18	3, 17	eqtr4d 2476	. 2 |- ( W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
19	1	str0 14208	. . 3 |- (/) = ( E ` (/) )
20		fvprc 5682	. . 3 |- ( -. W e. _V -> ( E ` W ) = (/) )
21		reldmsets 14192	. . . . 5 |- Rel dom sSet
22	21	ovprc1 6118	. . . 4 |- ( -. W e. _V -> ( W sSet <. D , C >. ) = (/) )
23	22	fveq2d 5692	. . 3 |- ( -. W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( E ` (/) ) )
24	19, 20, 23	3eqtr4a 2499	. 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 1364   e. wcel 1761   =/= wne 2604   _Vcvv 2970   \ cdif 3322   (/)c0 3634   {csn 3874   <.cop 3880   |` cres 4838   ` cfv 5415  (class class class)co 6090   ndxcnx 14167   sSet csts 14168  Slot cslot 14169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-res 4848  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-slot 14174  df-sets 14176

This theorem is referenced by:  resslem  14227  oppchomfval  14649  oppcbas  14653  rescbas  14738  rescco  14741  rescabs  14742  odubas  15299  oppglem  15858  mgplem  16586  opprlem  16710  sralem  17236  srasca  17240  sravsca  17241  opsrbaslem  17547  zlmlem  17848  zlmsca  17852  znbaslem  17871  thlbas  18021  thlle  18022  matbas  18214  matplusg  18215  matsca  18216  matvsca  18217  tuslem  19742  setsmsbas  19950  setsmsds  19951  tnglem  20126  tngds  20134  ttgval  23040  ttglem  23041  cchhllem  23052  resvlem  26219  zlmds  26313  zlmtset  26314  hlhilslem  35274