Theorem strsetsid 12449

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 12428	. . . 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 12430	. . . . . . . . 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 1006	. . . . . . 7 |- ( ph -> dom S C_ ( M ... N ) )
9	7	simp1d 1004	. . . . . . . . 9 |- ( ph -> ( M e. NN /\ N e. NN /\ M <_ N ) )
10	9	simp1d 1004	. . . . . . . 8 |- ( ph -> M e. NN )
11		fzssnn 10024	. . . . . . . 8 |- ( M e. NN -> ( M ... N ) C_ NN )
12	10, 11	syl 14	. . . . . . 7 |- ( ph -> ( M ... N ) C_ NN )
13	8, 12	sstrd 3157	. . . . . 6 |- ( ph -> dom S C_ NN )
14	13, 4	sseldd 3148	. . . . 5 |- ( ph -> ( E ` ndx ) e. NN )
15	5, 3, 14	strnfvnd 12436	. . . 4 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
16		strsetsid.f	. . . . 5 |- ( ph -> Fun S )
17		funfvex 5513	. . . . 5 |- ( ( Fun S /\ ( E ` ndx ) e. dom S ) -> ( S ` ( E ` ndx ) ) e. _V )
18	16, 4, 17	syl2anc 409	. . . 4 |- ( ph -> ( S ` ( E ` ndx ) ) e. _V )
19	15, 18	eqeltrd 2247	. . 3 |- ( ph -> ( E ` S ) e. _V )
20		setsvala 12447	. . 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 1233	. 2 |- ( ph -> ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) )
22	15	opeq2d 3772	. . . 4 |- ( ph -> <. ( E ` ndx ) , ( E ` S ) >. = <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. )
23	22	sneqd 3596	. . 3 |- ( ph -> { <. ( E ` ndx ) , ( E ` S ) >. } = { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } )
24	23	uneq2d 3281	. 2 |- ( ph -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( E ` S ) >. } ) = ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) )
25		nnssz 9229	. . . . 5 |- NN C_ ZZ
26	13, 25	sstrdi 3159	. . . 4 |- ( ph -> dom S C_ ZZ )
27		zdceq 9287	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> DECID x = y )
28	27	rgen2a 2524	. . . 4 |- A. x e. ZZ A. y e. ZZ DECID x = y
29		ssralv 3211	. . . . . 6 |- ( dom S C_ ZZ -> ( A. y e. ZZ DECID x = y -> A. y e. dom SDECID x = y ) )
30	29	ralimdv 2538	. . . . 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 3211	. . . . 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 1439	. . 3 |- ( ph -> A. x e. dom S A. y e. dom SDECID x = y )
34		funresdfunsndc 6485	. . 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 1233	. 2 |- ( ph -> ( ( S |` ( _V \ { ( E ` ndx ) } ) ) u. { <. ( E ` ndx ) , ( S ` ( E ` ndx ) ) >. } ) = S )
36	21, 24, 35	3eqtrrd 2208	1 |- ( ph -> S = ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) )

Syntax hints:   -> wi 4  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   \ cdif 3118   u. cun 3119   C_ wss 3121   (/)c0 3414   {csn 3583   <.cop 3586   class class class wbr 3989   dom cdm 4611   |` cres 4613   Fun wfun 5192   ` cfv 5198  (class class class)co 5853   <_ cle 7955   NNcn 8878   ZZcz 9212   ...cfz 9965   Struct cstr 12412   ndxcnx 12413   sSet csts 12414  Slot cslot 12415

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-struct 12418  df-slot 12420  df-sets 12423

This theorem is referenced by: (None)