Theorem setsslid 13123

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 13103	. . . 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 1359	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
5	4	fveq2d 5639	. 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 5051	. . . 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 4318	. . . . . 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 4272	. . . . 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 4538	. . . 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 597	. . 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 13092	. 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 3696	. . . . 5 |- ( ( E ` ndx ) e. NN -> ( E ` ndx ) e. { ( E ` ndx ) } )
18		fvres 5659	. . . . 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 5023	. . . . . . . . 9 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) )
21		incom 3397	. . . . . . . . . . . 12 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) )
22		disjdif 3565	. . . . . . . . . . . 12 |- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/)
23	21, 22	eqtri 2250	. . . . . . . . . . 11 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/)
24	23	reseq2i 5008	. . . . . . . . . 10 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) )
25		res0 5015	. . . . . . . . . 10 |- ( W |` (/) ) = (/)
26	24, 25	eqtri 2250	. . . . . . . . 9 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/)
27	20, 26	eqtri 2250	. . . . . . . 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 2813	. . . . . . . . . 10 |- ( E ` ndx ) e. _V
30	8	elexd 2814	. . . . . . . . . 10 |- ( ( W e. A /\ C e. V ) -> C e. _V )
31		opelxpi 4755	. . . . . . . . . 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 4827	. . . . . . . . . 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 5206	. . . . . . . . . 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 3279	. . . . . . . . 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 5049	. . . . . . . 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 3360	. . . . . 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 5025	. . . . . 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 3526	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. }
45		uncom 3349	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } )
46	44, 45	eqtr3i 2252	. . . . . 6 |- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } )
47	42, 43, 46	3eqtr4g 2287	. . . . 5 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
48	47	fveq1d 5637	. . . 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 2276	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
50		fvsng 5845	. . . 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 2262	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C )
53	5, 16, 52	3eqtrrd 2267	1 |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   \ cdif 3195   u. cun 3196   i^i cin 3197   C_ wss 3198   (/)c0 3492   {csn 3667   <.cop 3670   X. cxp 4721   dom cdm 4723   |` cres 4725   Rel wrel 4728   ` cfv 5324  (class class class)co 6013   NNcn 9133   ndxcnx 13069   sSet csts 13070  Slot cslot 13071

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-slot 13076  df-sets 13079

This theorem is referenced by:  ressbasd  13140  mgpplusgg  13927  opprmulfvalg  14073  rmodislmod  14355  srascag  14446  sravscag  14447  sraipg  14448  zlmsca  14636  zlmvscag  14637  znle  14641  setsmstsetg  15195  setsiedg  15893