Theorem setsnid 15915

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 15876	. . 3 |- ( W e. _V -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
4		ovex 6678	. . . . 5 |- ( W sSet <. D , C >. ) e. _V
5	4, 1	strfvn 15879	. . . 4 |- ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
6		setsres 15901	. . . . . 6 |- ( W e. _V -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
7	6	fveq1d 6193	. . . . 5 |- ( W e. _V -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
8		fvex 6201	. . . . . . 7 |- ( E ` ndx ) e. _V
9		setsnid.n	. . . . . . 7 |- ( E ` ndx ) =/= D
10		eldifsn 4317	. . . . . . 7 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
11	8, 9, 10	mpbir2an 955	. . . . . 6 |- ( E ` ndx ) e. ( _V \ { D } )
12		fvres 6207	. . . . . 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 6207	. . . . . 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 2679	. . . 4 |- ( W e. _V -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
17	5, 16	syl5eq 2668	. . 3 |- ( W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( W ` ( E ` ndx ) ) )
18	3, 17	eqtr4d 2659	. 2 |- ( W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
19	1	str0 15911	. . 3 |- (/) = ( E ` (/) )
20		fvprc 6185	. . 3 |- ( -. W e. _V -> ( E ` W ) = (/) )
21		reldmsets 15886	. . . . 5 |- Rel dom sSet
22	21	ovprc1 6684	. . . 4 |- ( -. W e. _V -> ( W sSet <. D , C >. ) = (/) )
23	22	fveq2d 6195	. . 3 |- ( -. W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( E ` (/) ) )
24	19, 20, 23	3eqtr4a 2682	. 2 |- ( -. W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
25	18, 24	pm2.61i 176	1 |- ( E ` W ) = ( E ` ( W sSet <. D , C >. ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   |` cres 5116   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855  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  ax-un 6949

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-ne 2795  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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-slot 15861  df-sets 15864

This theorem is referenced by:  resslem  15933  oppchomfval  16374  oppcbas  16378  rescbas  16489  rescco  16492  rescabs  16493  odubas  17133  oppglem  17780  mgplem  18494  opprlem  18628  rmodislmod  18931  sralem  19177  srasca  19181  sravsca  19182  opsrbaslem  19477  opsrbaslemOLD  19478  zlmlem  19865  zlmsca  19869  znbaslem  19886  znbaslemOLD  19887  thlbas  20040  thlle  20041  matbas  20219  matplusg  20220  matsca  20221  matvsca  20222  tuslem  22071  setsmsbas  22280  setsmsds  22281  tnglem  22444  tngds  22452  ttgval  25755  ttglem  25756  cchhllem  25767  setsvtx  25927  resvlem  29831  zlmds  30008  zlmtset  30009  hlhilslem  37230  cznrnglem  41953  cznabel  41954  cznrng  41955