Theorem setsslnid 13124

Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)

Hypotheses

Ref	Expression
setsslid.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
setsslnid.n	|- ( E ` ndx ) =/= D
setsslnid.d	|- D e. NN

Assertion

Ref	Expression
setsslnid	|- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Proof of Theorem setsslnid

Step	Hyp	Ref	Expression
1		setsslnid.d	. . . . 5 |- D e. NN
2		setsresg 13110	. . . . 5 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
3	1, 2	mp3an2 1359	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
4	3	fveq1d 5637	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
5		setsslid.e	. . . . . . 7 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
6	5	simpri 113	. . . . . 6 |- ( E ` ndx ) e. NN
7	6	elexi 2813	. . . . 5 |- ( E ` ndx ) e. _V
8		setsslnid.n	. . . . 5 |- ( E ` ndx ) =/= D
9		eldifsn 3798	. . . . 5 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
10	7, 8, 9	mpbir2an 948	. . . 4 |- ( E ` ndx ) e. ( _V \ { D } )
11		fvres 5659	. . . 4 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) )
12	10, 11	ax-mp 5	. . 3 |- ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
13		fvres 5659	. . . 4 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
14	10, 13	ax-mp 5	. . 3 |- ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) )
15	4, 12, 14	3eqtr3g 2285	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
16	5	simpli 111	. . 3 |- E = Slot ( E ` ndx )
17		setsex 13104	. . . 4 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( W sSet <. D , C >. ) e. _V )
18	1, 17	mp3an2 1359	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. D , C >. ) e. _V )
19	6	a1i 9	. . 3 |- ( ( W e. A /\ C e. V ) -> ( E ` ndx ) e. NN )
20	16, 18, 19	strnfvnd 13092	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) )
21		simpl 109	. . 3 |- ( ( W e. A /\ C e. V ) -> W e. A )
22	16, 21, 19	strnfvnd 13092	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
23	15, 20, 22	3eqtr4rd 2273	1 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2800   \ cdif 3195   {csn 3667   <.cop 3670   |` cres 4725   ` cfv 5324  (class class class)co 6013   NNcn 9133   ndxcnx 13069   sSet csts 13070  Slot cslot 13071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-slot 13076  df-sets 13079

This theorem is referenced by:  resseqnbasd  13146  mgpbasg  13929  mgpscag  13930  mgptsetg  13931  mgpdsg  13933  opprsllem  14077  rmodislmod  14355  sralemg  14442  srascag  14446  sravscag  14447  zlmlemg  14632  zlmsca  14636  znbaslemnn  14643  setsmsbasg  15193  setsmsdsg  15194  setsvtx  15892