Theorem setsnid 14760

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 14731	. . 3 |- ( W e. _V -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
4		ovex 6298	. . . . 5 |- ( W sSet <. D , C >. ) e. _V
5	4, 1	strfvn 14733	. . . 4 |- ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
6		setsres 14746	. . . . . 6 |- ( W e. _V -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
7	6	fveq1d 5850	. . . . 5 |- ( W e. _V -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
8		fvex 5858	. . . . . . 7 |- ( E ` ndx ) e. _V
9		setsnid.n	. . . . . . 7 |- ( E ` ndx ) =/= D
10		eldifsn 4141	. . . . . . 7 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
11	8, 9, 10	mpbir2an 918	. . . . . 6 |- ( E ` ndx ) e. ( _V \ { D } )
12		fvres 5862	. . . . . 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 5862	. . . . . 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 2518	. . . 4 |- ( W e. _V -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
17	5, 16	syl5eq 2507	. . 3 |- ( W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( W ` ( E ` ndx ) ) )
18	3, 17	eqtr4d 2498	. 2 |- ( W e. _V -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
19	1	str0 14756	. . 3 |- (/) = ( E ` (/) )
20		fvprc 5842	. . 3 |- ( -. W e. _V -> ( E ` W ) = (/) )
21		reldmsets 14739	. . . . 5 |- Rel dom sSet
22	21	ovprc1 6301	. . . 4 |- ( -. W e. _V -> ( W sSet <. D , C >. ) = (/) )
23	22	fveq2d 5852	. . 3 |- ( -. W e. _V -> ( E ` ( W sSet <. D , C >. ) ) = ( E ` (/) ) )
24	19, 20, 23	3eqtr4a 2521	. 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 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   \ cdif 3458   (/)c0 3783   {csn 4016   <.cop 4022   |` cres 4990   ` cfv 5570  (class class class)co 6270   ndxcnx 14713   sSet csts 14714  Slot cslot 14715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-slot 14720  df-sets 14722

This theorem is referenced by:  resslem  14776  oppchomfval  15202  oppcbas  15206  rescbas  15317  rescco  15320  rescabs  15321  odubas  15962  oppglem  16584  mgplem  17341  opprlem  17472  sralem  18018  srasca  18022  sravsca  18023  opsrbaslem  18337  zlmlem  18729  zlmsca  18733  znbaslem  18750  thlbas  18900  thlle  18901  matbas  19082  matplusg  19083  matsca  19084  matvsca  19085  tuslem  20936  setsmsbas  21144  setsmsds  21145  tnglem  21320  tngds  21328  ttgval  24380  ttglem  24381  cchhllem  24392  resvlem  28056  zlmds  28179  zlmtset  28180  uhgrepe  32750  cznrnglem  33015  cznabel  33016  cznrng  33017  hlhilslem  38065