Theorem strsetsid 13105

Description: Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strsetsid.e	|- E = Slot ( E ` ndx )
strsetsid.s	|- ( ph -> S Struct <. M , N >. )
strsetsid.f	|- ( ph -> Fun S )
strsetsid.d	|- ( ph -> ( E ` ndx ) e. dom S )

Assertion

Ref	Expression
strsetsid	|- ( ph -> S = ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) )

Proof of Theorem strsetsid

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		strsetsid.s	. . . 4 |- ( ph -> S Struct <. M , N >. )
2		structex 13084	. . . 4 |- ( S Struct <. M , N >. -> S e. _V )
3	1, 2	syl 14	. . 3 |- ( ph -> S e. _V )
4		strsetsid.d	. . 3 |- ( ph -> ( E ` ndx ) e. dom S )
5		strsetsid.e	. . . . 5 |- E = Slot ( E ` ndx )
6		isstructim 13086	. . . . . . . . 9 |- ( S Struct <. M , N >. -> ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ Fun ( S \ { (/) } ) /\ dom S C_ ( M ... N ) ) )
7	1, 6	syl 14	. . . . . . . 8 |- ( ph -> ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ Fun ( S \ { (/) } ) /\ dom S C_ ( M ... N ) ) )
8	7	simp3d 1035	. . . . . . 7 |- ( ph -> dom S C_ ( M ... N ) )
9	7	simp1d 1033	. . . . . . . . 9 |- ( ph -> ( M e. NN /\ N e. NN /\ M <_ N ) )
10	9	simp1d 1033	. . . . . . . 8 |- ( ph -> M e. NN )
11		fzssnn 10293	. . . . . . . 8 |- ( M e. NN -> ( M ... N ) C_ NN )
12	10, 11	syl 14	. . . . . . 7 |- ( ph -> ( M ... N ) C_ NN )
13	8, 12	sstrd 3235	. . . . . 6 |- ( ph -> dom S C_ NN )
14	13, 4	sseldd 3226	. . . . 5 |- ( ph -> ( E ` ndx ) e. NN )
15	5, 3, 14	strnfvnd 13092	. . . 4 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
16		strsetsid.f	. . . . 5 |- ( ph -> Fun S )
17		funfvex 5652	. . . . 5 |- ( ( Fun S /\ ( E ` ndx ) e. dom S ) -> ( S ` ( E ` ndx ) ) e. _V )
18	16, 4, 17	syl2anc 411	. . . 4 |- ( ph -> ( S ` ( E ` ndx ) ) e. _V )
19	15, 18	eqeltrd 2306	. . 3 |- ( ph -> ( E ` S ) e. _V )
20		setsvala 13103	. . 3 |- ( ( S e. _V /\ ( E ` ndx ) e. dom S /\ ( E ` S ) e. _V ) -> ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) )
21	3, 4, 19, 20	syl3anc 1271	. 2 |- ( ph -> ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) )
22	15	opeq2d 3867	. . . 4 |- ( ph -> <. ( E ` ndx ) , ( E ` S ) >. = <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. )
23	22	sneqd 3680	. . 3 |- ( ph -> { <. ( E ` ndx ) , ( E ` S ) >. } = { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } )
24	23	uneq2d 3359	. 2 |- ( ph -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) )
25		nnssz 9486	. . . . 5 |- NN C_ ZZ
26	13, 25	sstrdi 3237	. . . 4 |- ( ph -> dom S C_ ZZ )
27		zdceq 9545	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> DECID x = y )
28	27	rgen2a 2584	. . . 4 |- A. x e. ZZ A. y e. ZZ DECID x = y
29		ssralv 3289	. . . . . 6 |- ( dom S C_ ZZ -> ( A. y e. ZZ DECID x = y -> A. y e. dom SDECID x = y ) )
30	29	ralimdv 2598	. . . . 5 |- ( dom S C_ ZZ -> ( A. x e. ZZ A. y e. ZZ DECID x = y -> A. x e. ZZ A. y e. dom SDECID x = y ) )
31		ssralv 3289	. . . . 5 |- ( dom S C_ ZZ -> ( A. x e. ZZ A. y e. dom SDECID x = y -> A. x e. dom S A. y e. dom SDECID x = y ) )
32	30, 31	syld 45	. . . 4 |- ( dom S C_ ZZ -> ( A. x e. ZZ A. y e. ZZ DECID x = y -> A. x e. dom S A. y e. dom SDECID x = y ) )
33	26, 28, 32	mpisyl 1489	. . 3 |- ( ph -> A. x e. dom S A. y e. dom SDECID x = y )
34		funresdfunsndc 6669	. . 3 |- ( ( A. x e. dom S A. y e. dom SDECID x = y /\ Fun S /\ ( E ` ndx ) e. dom S ) -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) = S )
35	33, 16, 4, 34	syl3anc 1271	. 2 |- ( ph -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) = S )
36	21, 24, 35	3eqtrrd 2267	1 |- ( ph -> S = ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) )

Syntax hints:   -> wi 4  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2800   \ cdif 3195   u. cun 3196   C_ wss 3198   (/)c0 3492   {csn 3667   <.cop 3670   class class class wbr 4086   dom cdm 4723   |` cres 4725   Fun wfun 5318   ` cfv 5324  (class class class)co 6013   <_ cle 8205   NNcn 9133   ZZcz 9469   ...cfz 10233   Struct cstr 13068   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  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  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-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  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-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-struct 13074  df-slot 13076  df-sets 13079

This theorem is referenced by:  strressid  13144