Theorem bassetsnn 13132

Description: The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
basprssdmsets.s	|- ( ph -> S Struct X )
bassetsnn.i	|- ( ph -> I e. NN )
basprssdmsets.w	|- ( ph -> E e. W )
basprssdmsets.b	|- ( ph -> ( Base ` ndx ) e. dom S )

Assertion

Ref	Expression
bassetsnn	|- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Proof of Theorem bassetsnn

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ ( Base ` ndx ) = I ) -> ( Base ` ndx ) = I )
2		bassetsnn.i	. . . . . . . . . 10 |- ( ph -> I e. NN )
3		snidg 3696	. . . . . . . . . 10 |- ( I e. NN -> I e. { I } )
4	2, 3	syl 14	. . . . . . . . 9 |- ( ph -> I e. { I } )
5		basprssdmsets.w	. . . . . . . . . 10 |- ( ph -> E e. W )
6		dmsnopg 5206	. . . . . . . . . 10 |- ( E e. W -> dom { <. I , E >. } = { I } )
7	5, 6	syl 14	. . . . . . . . 9 |- ( ph -> dom { <. I , E >. } = { I } )
8	4, 7	eleqtrrd 2309	. . . . . . . 8 |- ( ph -> I e. dom { <. I , E >. } )
9		elun2 3373	. . . . . . . 8 |- ( I e. dom { <. I , E >. } -> I e. ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> I e. ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } ) )
11		dmun 4936	. . . . . . 7 |- dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) = ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } )
12	10, 11	eleqtrrdi 2323	. . . . . 6 |- ( ph -> I e. dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
13		basprssdmsets.s	. . . . . . . . 9 |- ( ph -> S Struct X )
14		structex 13087	. . . . . . . . 9 |- ( S Struct X -> S e. _V )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> S e. _V )
16		opexg 4318	. . . . . . . . 9 |- ( ( I e. NN /\ E e. W ) -> <. I , E >. e. _V )
17	2, 5, 16	syl2anc 411	. . . . . . . 8 |- ( ph -> <. I , E >. e. _V )
18		setsvalg 13105	. . . . . . . 8 |- ( ( S e. _V /\ <. I , E >. e. _V ) -> ( S sSet <. I , E >. ) = ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
19	15, 17, 18	syl2anc 411	. . . . . . 7 |- ( ph -> ( S sSet <. I , E >. ) = ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
20	19	dmeqd 4931	. . . . . 6 |- ( ph -> dom ( S sSet <. I , E >. ) = dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
21	12, 20	eleqtrrd 2309	. . . . 5 |- ( ph -> I e. dom ( S sSet <. I , E >. ) )
22	21	adantr 276	. . . 4 |- ( ( ph /\ ( Base ` ndx ) = I ) -> I e. dom ( S sSet <. I , E >. ) )
23	1, 22	eqeltrd 2306	. . 3 |- ( ( ph /\ ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
24		basendxnn 13131	. . . . . . . . . . 11 |- ( Base ` ndx ) e. NN
25	24	elexi 2813	. . . . . . . . . 10 |- ( Base ` ndx ) e. _V
26	25	a1i 9	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. _V )
27		simpr 110	. . . . . . . . . . . 12 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) e. dom { <. I , E >. } )
28	7	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> dom { <. I , E >. } = { I } )
29	27, 28	eleqtrd 2308	. . . . . . . . . . 11 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) e. { I } )
30		elsni 3685	. . . . . . . . . . 11 |- ( ( Base ` ndx ) e. { I } -> ( Base ` ndx ) = I )
31	29, 30	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) = I )
32	31	stoic1a 1469	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> -. ( Base ` ndx ) e. dom { <. I , E >. } )
33	26, 32	eldifd 3208	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( _V \ dom { <. I , E >. } ) )
34		basprssdmsets.b	. . . . . . . . 9 |- ( ph -> ( Base ` ndx ) e. dom S )
35	34	adantr 276	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom S )
36	33, 35	elind 3390	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( ( _V \ dom { <. I , E >. } ) i^i dom S ) )
37		dmres 5032	. . . . . . 7 |- dom ( S |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom S )
38	36, 37	eleqtrrdi 2323	. . . . . 6 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S |` ( _V \ dom { <. I , E >. } ) ) )
39		elun1 3372	. . . . . 6 |- ( ( Base ` ndx ) e. dom ( S |` ( _V \ dom { <. I , E >. } ) ) -> ( Base ` ndx ) e. ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } ) )
40	38, 39	syl 14	. . . . 5 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } ) )
41	40, 11	eleqtrrdi 2323	. . . 4 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
42	20	adantr 276	. . . 4 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> dom ( S sSet <. I , E >. ) = dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
43	41, 42	eleqtrrd 2309	. . 3 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
44	24	nnzi 9493	. . . . 5 |- ( Base ` ndx ) e. ZZ
45	2	nnzd 9594	. . . . 5 |- ( ph -> I e. ZZ )
46		zdceq 9548	. . . . 5 |- ( ( ( Base ` ndx ) e. ZZ /\ I e. ZZ ) -> DECID ( Base ` ndx ) = I )
47	44, 45, 46	sylancr 414	. . . 4 |- ( ph -> DECID ( Base ` ndx ) = I )
48		exmiddc 841	. . . 4 |- (DECID ( Base ` ndx ) = I -> ( ( Base ` ndx ) = I \/ -. ( Base ` ndx ) = I ) )
49	47, 48	syl 14	. . 3 |- ( ph -> ( ( Base ` ndx ) = I \/ -. ( Base ` ndx ) = I ) )
50	23, 43, 49	mpjaodan 803	. 2 |- ( ph -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
51	50, 21	prssd 3830	1 |- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   _Vcvv 2800   \ cdif 3195   u. cun 3196   i^i cin 3197   C_ wss 3198   {csn 3667   {cpr 3668   <.cop 3670   class class class wbr 4086   dom cdm 4723   |` cres 4725   ` cfv 5324  (class class class)co 6013   NNcn 9136   ZZcz 9472   Struct cstr 13071   ndxcnx 13072   sSet csts 13073   Basecbs 13075

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 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-ltadd 8141

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-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-z 9473  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082

This theorem is referenced by:  setsvtx  15895  setsiedg  15896