Theorem setsslnid 13099

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 13085	. . . . 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 5631	. . 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 2812	. . . . 5 |- ( E ` ndx ) e. _V
8		setsslnid.n	. . . . 5 |- ( E ` ndx ) =/= D
9		eldifsn 3795	. . . . 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 5653	. . . 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 5653	. . . 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 13079	. . . 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 13067	. 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 13067	. 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 2799   \ cdif 3194   {csn 3666   <.cop 3669   |` cres 4721   ` cfv 5318  (class class class)co 6007   NNcn 9121   ndxcnx 13044   sSet csts 13045  Slot cslot 13046

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-slot 13051  df-sets 13054

This theorem is referenced by:  resseqnbasd  13121  mgpbasg  13904  mgpscag  13905  mgptsetg  13906  mgpdsg  13908  opprsllem  14052  rmodislmod  14330  sralemg  14417  srascag  14421  sravscag  14422  zlmlemg  14607  zlmsca  14611  znbaslemnn  14618  setsmsbasg  15168  setsmsdsg  15169  setsvtx  15867