Theorem setsslid 12669

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 12649	. . . 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 1336	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
5	4	fveq2d 5558	. 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 4982	. . . 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 4257	. . . . . 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 4213	. . . . 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 4474	. . . 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 12638	. 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 3647	. . . . 5 |- ( ( E ` ndx ) e. NN -> ( E ` ndx ) e. { ( E ` ndx ) } )
18		fvres 5578	. . . . 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 4954	. . . . . . . . 9 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) )
21		incom 3351	. . . . . . . . . . . 12 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) )
22		disjdif 3519	. . . . . . . . . . . 12 |- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/)
23	21, 22	eqtri 2214	. . . . . . . . . . 11 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/)
24	23	reseq2i 4939	. . . . . . . . . 10 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) )
25		res0 4946	. . . . . . . . . 10 |- ( W |` (/) ) = (/)
26	24, 25	eqtri 2214	. . . . . . . . 9 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/)
27	20, 26	eqtri 2214	. . . . . . . 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 2772	. . . . . . . . . 10 |- ( E ` ndx ) e. _V
30	8	elexd 2773	. . . . . . . . . 10 |- ( ( W e. A /\ C e. V ) -> C e. _V )
31		opelxpi 4691	. . . . . . . . . 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 4762	. . . . . . . . . 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 5137	. . . . . . . . . 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 3233	. . . . . . . . 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 4980	. . . . . . . 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 3314	. . . . . 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 4956	. . . . . 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 3480	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. }
45		uncom 3303	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } )
46	44, 45	eqtr3i 2216	. . . . . 6 |- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } )
47	42, 43, 46	3eqtr4g 2251	. . . . 5 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
48	47	fveq1d 5556	. . . 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 2240	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
50		fvsng 5754	. . . 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 2226	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C )
53	5, 16, 52	3eqtrrd 2231	1 |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   \ cdif 3150   u. cun 3151   i^i cin 3152   C_ wss 3153   (/)c0 3446   {csn 3618   <.cop 3621   X. cxp 4657   dom cdm 4659   |` cres 4661   Rel wrel 4664   ` cfv 5254  (class class class)co 5918   NNcn 8982   ndxcnx 12615   sSet csts 12616  Slot cslot 12617

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-slot 12622  df-sets 12625

This theorem is referenced by:  ressbasd  12685  mgpplusgg  13420  opprmulfvalg  13566  rmodislmod  13847  srascag  13938  sravscag  13939  sraipg  13940  zlmsca  14120  zlmvscag  14121  znle  14125  setsmstsetg  14649