Theorem setsslnid 13348

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 13334	. . . . 5 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
3	1, 2	mp3an2 1362	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
4	3	fveq1d 5677	. . 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 2828	. . . . 5 |- ( E ` ndx ) e. _V
8		setsslnid.n	. . . . 5 |- ( E ` ndx ) =/= D
9		eldifsn 3825	. . . . 5 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
10	7, 8, 9	mpbir2an 951	. . . 4 |- ( E ` ndx ) e. ( _V \ { D } )
11		fvres 5699	. . . 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 5699	. . . 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 2290	. 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 13328	. . . 4 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( W sSet <. D , C >. ) e. _V )
18	1, 17	mp3an2 1362	. . 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 13316	. 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 13316	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
23	15, 20, 22	3eqtr4rd 2278	1 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   \ cdif 3211   {csn 3694   <.cop 3697   |` cres 4756   ` cfv 5357  (class class class)co 6058   NNcn 9254   ndxcnx 13293   sSet csts 13294  Slot cslot 13295

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-slot 13300  df-sets 13303

This theorem is referenced by:  resseqnbasd  13370  mgpbasg  14165  mgpscag  14166  mgptsetg  14167  mgpdsg  14169  opprsllem  14317  rmodislmod  14625  sralemg  14712  srascag  14716  sravscag  14717  zlmlemg  14902  zlmsca  14906  znbaslemnn  14913  setsmsbasg  15470  setsmsdsg  15471  setsvtx  16172