Theorem setsslid 12527

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

Hypothesis

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

Assertion

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

Proof of Theorem setsslid

Step	Hyp	Ref	Expression
1		setsslid.e	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2	1	simpri 113	. . . 4 |- ( E ` ndx ) e. NN
3		setsvala 12507	. . . 4 |- ( ( W e. A /\ ( E ` ndx ) e. NN /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
4	2, 3	mp3an2 1335	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
5	4	fveq2d 5531	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) = ( E ` ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) )
6	1	simpli 111	. . 3 |- E = Slot ( E ` ndx )
7		resexg 4959	. . . 4 |- ( W e. A -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V )
8		simpr 110	. . . . . 6 |- ( ( W e. A /\ C e. V ) -> C e. V )
9		opexg 4240	. . . . . 6 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
10	2, 8, 9	sylancr 414	. . . . 5 |- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
11		snexg 4196	. . . . 5 |- ( <. ( E ` ndx ) , C >. e. _V -> { <. ( E ` ndx ) , C >. } e. _V )
12	10, 11	syl 14	. . . 4 |- ( ( W e. A /\ C e. V ) -> { <. ( E ` ndx ) , C >. } e. _V )
13		unexg 4455	. . . 4 |- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V /\ { <. ( E ` ndx ) , C >. } e. _V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V )
14	7, 12, 13	syl2an2r 595	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V )
15	2	a1i 9	. . 3 |- ( ( W e. A /\ C e. V ) -> ( E ` ndx ) e. NN )
16	6, 14, 15	strnfvnd 12496	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) )
17		snidg 3633	. . . . 5 |- ( ( E ` ndx ) e. NN -> ( E ` ndx ) e. { ( E ` ndx ) } )
18		fvres 5551	. . . . 5 |- ( ( E ` ndx ) e. { ( E ` ndx ) } -> ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) )
19	2, 17, 18	mp2b 8	. . . 4 |- ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) )
20		resres 4931	. . . . . . . . 9 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) )
21		incom 3339	. . . . . . . . . . . 12 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) )
22		disjdif 3507	. . . . . . . . . . . 12 |- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/)
23	21, 22	eqtri 2208	. . . . . . . . . . 11 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/)
24	23	reseq2i 4916	. . . . . . . . . 10 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) )
25		res0 4923	. . . . . . . . . 10 |- ( W |` (/) ) = (/)
26	24, 25	eqtri 2208	. . . . . . . . 9 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/)
27	20, 26	eqtri 2208	. . . . . . . 8 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/)
28	27	a1i 9	. . . . . . 7 |- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/) )
29	2	elexi 2761	. . . . . . . . . 10 |- ( E ` ndx ) e. _V
30	8	elexd 2762	. . . . . . . . . 10 |- ( ( W e. A /\ C e. V ) -> C e. _V )
31		opelxpi 4670	. . . . . . . . . 10 |- ( ( ( E ` ndx ) e. _V /\ C e. _V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
32	29, 30, 31	sylancr 414	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
33		relsng 4741	. . . . . . . . . 10 |- ( <. ( E ` ndx ) , C >. e. _V -> ( Rel { <. ( E ` ndx ) , C >. } <-> <. ( E ` ndx ) , C >. e. ( _V X. _V ) ) )
34	10, 33	syl 14	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> ( Rel { <. ( E ` ndx ) , C >. } <-> <. ( E ` ndx ) , C >. e. ( _V X. _V ) ) )
35	32, 34	mpbird 167	. . . . . . . 8 |- ( ( W e. A /\ C e. V ) -> Rel { <. ( E ` ndx ) , C >. } )
36		dmsnopg 5112	. . . . . . . . . 10 |- ( C e. V -> dom { <. ( E ` ndx ) , C >. } = { ( E ` ndx ) } )
37	36	adantl 277	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> dom { <. ( E ` ndx ) , C >. } = { ( E ` ndx ) } )
38		eqimss 3221	. . . . . . . . 9 |- ( dom { <. ( E ` ndx ) , C >. } = { ( E ` ndx ) } -> dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } )
39	37, 38	syl 14	. . . . . . . 8 |- ( ( W e. A /\ C e. V ) -> dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } )
40		relssres 4957	. . . . . . . 8 |- ( ( Rel { <. ( E ` ndx ) , C >. } /\ dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
41	35, 39, 40	syl2anc 411	. . . . . . 7 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
42	28, 41	uneq12d 3302	. . . . . 6 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) ) = ( (/) u. { <. ( E ` ndx ) , C >. } ) )
43		resundir 4933	. . . . . 6 |- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) )
44		un0 3468	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. }
45		uncom 3291	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } )
46	44, 45	eqtr3i 2210	. . . . . 6 |- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } )
47	42, 43, 46	3eqtr4g 2245	. . . . 5 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
48	47	fveq1d 5529	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
49	19, 48	eqtr3id 2234	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
50		fvsng 5725	. . . 4 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
51	2, 8, 50	sylancr 414	. . 3 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
52	49, 51	eqtrd 2220	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C )
53	5, 16, 52	3eqtrrd 2225	1 |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   _Vcvv 2749   \ cdif 3138   u. cun 3139   i^i cin 3140   C_ wss 3141   (/)c0 3434   {csn 3604   <.cop 3607   X. cxp 4636   dom cdm 4638   |` cres 4640   Rel wrel 4643   ` cfv 5228  (class class class)co 5888   NNcn 8933   ndxcnx 12473   sSet csts 12474  Slot cslot 12475

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-slot 12480  df-sets 12483

This theorem is referenced by:  ressbasd  12541  mgpplusgg  13176  opprmulfvalg  13318  rmodislmod  13540  srascag  13631  sravscag  13632  sraipg  13633  zlmsca  13790  zlmvscag  13791  setsmstsetg  14277