Theorem setsslnid 13264

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 13250	. . . . 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 5672	. . 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 2826	. . . . 5 |- ( E ` ndx ) e. _V
8		setsslnid.n	. . . . 5 |- ( E ` ndx ) =/= D
9		eldifsn 3820	. . . . 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 5694	. . . 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 5694	. . . 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 2288	. 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 13244	. . . 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 13232	. 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 13232	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
23	15, 20, 22	3eqtr4rd 2276	1 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2813   \ cdif 3208   {csn 3689   <.cop 3692   |` cres 4751   ` cfv 5352  (class class class)co 6050   NNcn 9237   ndxcnx 13209   sSet csts 13210  Slot cslot 13211

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-slot 13216  df-sets 13219

This theorem is referenced by:  resseqnbasd  13286  mgpbasg  14070  mgpscag  14071  mgptsetg  14072  mgpdsg  14074  opprsllem  14218  rmodislmod  14499  sralemg  14586  srascag  14590  sravscag  14591  zlmlemg  14776  zlmsca  14780  znbaslemnn  14787  setsmsbasg  15344  setsmsdsg  15345  setsvtx  16046