Theorem strsetsid 13060

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 13039	. . . 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 13041	. . . . . . . . 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 10260	. . . . . . . 8 |- ( M e. NN -> ( M ... N ) C_ NN )
12	10, 11	syl 14	. . . . . . 7 |- ( ph -> ( M ... N ) C_ NN )
13	8, 12	sstrd 3234	. . . . . 6 |- ( ph -> dom S C_ NN )
14	13, 4	sseldd 3225	. . . . 5 |- ( ph -> ( E ` ndx ) e. NN )
15	5, 3, 14	strnfvnd 13047	. . . 4 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
16		strsetsid.f	. . . . 5 |- ( ph -> Fun S )
17		funfvex 5643	. . . . 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 13058	. . 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 3863	. . . 4 |- ( ph -> <. ( E ` ndx ) , ( E ` S ) >. = <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. )
23	22	sneqd 3679	. . 3 |- ( ph -> { <. ( E ` ndx ) , ( E ` S ) >. } = { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } )
24	23	uneq2d 3358	. 2 |- ( ph -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) )
25		nnssz 9459	. . . . 5 |- NN C_ ZZ
26	13, 25	sstrdi 3236	. . . 4 |- ( ph -> dom S C_ ZZ )
27		zdceq 9518	. . . . 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 3288	. . . . . 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 3288	. . . . 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 6650	. . 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 2799   \ cdif 3194   u. cun 3195   C_ wss 3197   (/)c0 3491   {csn 3666   <.cop 3669   class class class wbr 4082   dom cdm 4718   |` cres 4720   Fun wfun 5311   ` cfv 5317  (class class class)co 6000   <_ cle 8178   NNcn 9106   ZZcz 9442   ...cfz 10200   Struct cstr 13023   ndxcnx 13024   sSet csts 13025  Slot cslot 13026

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-struct 13029  df-slot 13031  df-sets 13034

This theorem is referenced by:  strressid  13099