Theorem setsslid 12009

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 112	. . . 4 |- ( E ` ndx ) e. NN
3		setsvala 11990	. . . 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 1303	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
5	4	fveq2d 5425	. 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 110	. . 3 |- E = Slot ( E ` ndx )
7		resexg 4859	. . . 4 |- ( W e. A -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V )
8		simpr 109	. . . . . 6 |- ( ( W e. A /\ C e. V ) -> C e. V )
9		opexg 4150	. . . . . 6 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
10	2, 8, 9	sylancr 410	. . . . 5 |- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
11		snexg 4108	. . . . 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 4364	. . . 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 584	. . 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 11979	. 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 3554	. . . . 5 |- ( ( E ` ndx ) e. NN -> ( E ` ndx ) e. { ( E ` ndx ) } )
18		fvres 5445	. . . . 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 4831	. . . . . . . . 9 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) )
21		incom 3268	. . . . . . . . . . . 12 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) )
22		disjdif 3435	. . . . . . . . . . . 12 |- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/)
23	21, 22	eqtri 2160	. . . . . . . . . . 11 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/)
24	23	reseq2i 4816	. . . . . . . . . 10 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) )
25		res0 4823	. . . . . . . . . 10 |- ( W |` (/) ) = (/)
26	24, 25	eqtri 2160	. . . . . . . . 9 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/)
27	20, 26	eqtri 2160	. . . . . . . 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 2698	. . . . . . . . . 10 |- ( E ` ndx ) e. _V
30	8	elexd 2699	. . . . . . . . . 10 |- ( ( W e. A /\ C e. V ) -> C e. _V )
31		opelxpi 4571	. . . . . . . . . 10 |- ( ( ( E ` ndx ) e. _V /\ C e. _V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
32	29, 30, 31	sylancr 410	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
33		relsng 4642	. . . . . . . . . 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 166	. . . . . . . 8 |- ( ( W e. A /\ C e. V ) -> Rel { <. ( E ` ndx ) , C >. } )
36		dmsnopg 5010	. . . . . . . . . 10 |- ( C e. V -> dom { <. ( E ` ndx ) , C >. } = { ( E ` ndx ) } )
37	36	adantl 275	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> dom { <. ( E ` ndx ) , C >. } = { ( E ` ndx ) } )
38		eqimss 3151	. . . . . . . . 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 4857	. . . . . . . 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 408	. . . . . . 7 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
42	28, 41	uneq12d 3231	. . . . . 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 4833	. . . . . 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 3396	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. }
45		uncom 3220	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } )
46	44, 45	eqtr3i 2162	. . . . . 6 |- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } )
47	42, 43, 46	3eqtr4g 2197	. . . . 5 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
48	47	fveq1d 5423	. . . 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	syl5eqr 2186	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
50		fvsng 5616	. . . 4 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
51	2, 8, 50	sylancr 410	. . 3 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
52	49, 51	eqtrd 2172	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C )
53	5, 16, 52	3eqtrrd 2177	1 |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   \ cdif 3068   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   {csn 3527   <.cop 3530   X. cxp 4537   dom cdm 4539   |` cres 4541   Rel wrel 4544   ` cfv 5123  (class class class)co 5774   NNcn 8720   ndxcnx 11956   sSet csts 11957  Slot cslot 11958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-slot 11963  df-sets 11966

This theorem is referenced by:  setsmstsetg  12650