Theorem setsid 15914

Description: Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypothesis

Ref	Expression
setsid.e	|- E = Slot ( E ` ndx )

Assertion

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

Proof of Theorem setsid

Step	Hyp	Ref	Expression
1		setsval 15888	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. ( E ` ndx ) , C >. ) = ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) )
2	1	fveq2d 6195	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) = ( E ` ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ) )
3		setsid.e	. . 3 |- E = Slot ( E ` ndx )
4		resexg 5442	. . . . 5 |- ( W e. A -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V )
5	4	adantr 481	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V )
6		snex 4908	. . . 4 |- { <. ( E ` ndx ) , C >. } e. _V
7		unexg 6959	. . . 4 |- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) e. _V /\ { <. ( E ` ndx ) , C >. } e. _V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V )
8	5, 6, 7	sylancl 694	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) e. _V )
9	3, 8	strfvnd 15876	. 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 ) ) )
10		fvex 6201	. . . . . 6 |- ( E ` ndx ) e. _V
11	10	snid 4208	. . . . 5 |- ( E ` ndx ) e. { ( E ` ndx ) }
12		fvres 6207	. . . . 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 ) ) )
13	11, 12	ax-mp 5	. . . 4 |- ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) )
14		resres 5409	. . . . . . . . 9 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) )
15		incom 3805	. . . . . . . . . . . 12 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) )
16		disjdif 4040	. . . . . . . . . . . 12 |- ( { ( E ` ndx ) } i^i ( _V \ { ( E ` ndx ) } ) ) = (/)
17	15, 16	eqtri 2644	. . . . . . . . . . 11 |- ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) = (/)
18	17	reseq2i 5393	. . . . . . . . . 10 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = ( W |` (/) )
19		res0 5400	. . . . . . . . . 10 |- ( W |` (/) ) = (/)
20	18, 19	eqtri 2644	. . . . . . . . 9 |- ( W |` ( ( _V \ { ( E ` ndx ) } ) i^i { ( E ` ndx ) } ) ) = (/)
21	14, 20	eqtri 2644	. . . . . . . 8 |- ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/)
22	21	a1i 11	. . . . . . 7 |- ( ( W e. A /\ C e. V ) -> ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) = (/) )
23		elex 3212	. . . . . . . . . . 11 |- ( C e. V -> C e. _V )
24	23	adantl 482	. . . . . . . . . 10 |- ( ( W e. A /\ C e. V ) -> C e. _V )
25		opelxpi 5148	. . . . . . . . . 10 |- ( ( ( E ` ndx ) e. _V /\ C e. _V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
26	10, 24, 25	sylancr 695	. . . . . . . . 9 |- ( ( W e. A /\ C e. V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
27		opex 4932	. . . . . . . . . 10 |- <. ( E ` ndx ) , C >. e. _V
28	27	relsn 5223	. . . . . . . . 9 |- ( Rel { <. ( E ` ndx ) , C >. } <-> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
29	26, 28	sylibr 224	. . . . . . . 8 |- ( ( W e. A /\ C e. V ) -> Rel { <. ( E ` ndx ) , C >. } )
30		dmsnopss 5607	. . . . . . . 8 |- dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) }
31		relssres 5437	. . . . . . . 8 |- ( ( Rel { <. ( E ` ndx ) , C >. } /\ dom { <. ( E ` ndx ) , C >. } C_ { ( E ` ndx ) } ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
32	29, 30, 31	sylancl 694	. . . . . . 7 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
33	22, 32	uneq12d 3768	. . . . . 6 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) ) = ( (/) u. { <. ( E ` ndx ) , C >. } ) )
34		resundir 5411	. . . . . 6 |- ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) |` { ( E ` ndx ) } ) u. ( { <. ( E ` ndx ) , C >. } |` { ( E ` ndx ) } ) )
35		un0 3967	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = { <. ( E ` ndx ) , C >. }
36		uncom 3757	. . . . . . 7 |- ( { <. ( E ` ndx ) , C >. } u. (/) ) = ( (/) u. { <. ( E ` ndx ) , C >. } )
37	35, 36	eqtr3i 2646	. . . . . 6 |- { <. ( E ` ndx ) , C >. } = ( (/) u. { <. ( E ` ndx ) , C >. } )
38	33, 34, 37	3eqtr4g 2681	. . . . 5 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) = { <. ( E ` ndx ) , C >. } )
39	38	fveq1d 6193	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) |` { ( E ` ndx ) } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
40	13, 39	syl5eqr 2670	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) )
41	10	a1i 11	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( E ` ndx ) e. _V )
42		fvsng 6447	. . . 4 |- ( ( ( E ` ndx ) e. _V /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
43	41, 42	sylancom 701	. . 3 |- ( ( W e. A /\ C e. V ) -> ( { <. ( E ` ndx ) , C >. } ` ( E ` ndx ) ) = C )
44	40, 43	eqtrd 2656	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( ( W |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , C >. } ) ` ( E ` ndx ) ) = C )
45	2, 9, 44	3eqtrrd 2661	1 |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   X. cxp 5112   dom cdm 5114   |` cres 5116   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-slot 15861  df-sets 15864

This theorem is referenced by:  ressbas  15930  oppchomfval  16374  oppccofval  16376  reschom  16490  oduleval  17131  oppgplusfval  17778  mgpplusg  18493  opprmulfval  18625  rmodislmod  18931  srasca  19181  sravsca  19182  sraip  19183  opsrle  19475  zlmsca  19869  zlmvsca  19870  znle  19884  thloc  20043  matmulr  20244  tuslem  22071  setsmstset  22282  tngds  22452  tngtset  22453  ttgval  25755  setsiedg  25928  resvsca  29830  hlhilnvl  37242  cznrng  41955  cznnring  41956