Theorem bassetsnn 13162

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 3699	. . . . . . . . . 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 5210	. . . . . . . . . 10 |- ( E e. W -> dom { <. I , E >. } = { I } )
7	5, 6	syl 14	. . . . . . . . 9 |- ( ph -> dom { <. I , E >. } = { I } )
8	4, 7	eleqtrrd 2310	. . . . . . . 8 |- ( ph -> I e. dom { <. I , E >. } )
9		elun2 3374	. . . . . . . 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 4940	. . . . . . 7 |- dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) = ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } )
12	10, 11	eleqtrrdi 2324	. . . . . 6 |- ( ph -> I e. dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
13		basprssdmsets.s	. . . . . . . . 9 |- ( ph -> S Struct X )
14		structex 13117	. . . . . . . . 9 |- ( S Struct X -> S e. _V )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> S e. _V )
16		opexg 4322	. . . . . . . . 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 13135	. . . . . . . 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 4935	. . . . . 6 |- ( ph -> dom ( S sSet <. I , E >. ) = dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
21	12, 20	eleqtrrd 2310	. . . . 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 2307	. . 3 |- ( ( ph /\ ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
24		basendxnn 13161	. . . . . . . . . . 11 |- ( Base ` ndx ) e. NN
25	24	elexi 2814	. . . . . . . . . 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 2309	. . . . . . . . . . 11 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) e. { I } )
30		elsni 3688	. . . . . . . . . . 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 1471	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> -. ( Base ` ndx ) e. dom { <. I , E >. } )
33	26, 32	eldifd 3209	. . . . . . . 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 3391	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( ( _V \ dom { <. I , E >. } ) i^i dom S ) )
37		dmres 5036	. . . . . . 7 |- dom ( S |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom S )
38	36, 37	eleqtrrdi 2324	. . . . . 6 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S |` ( _V \ dom { <. I , E >. } ) ) )
39		elun1 3373	. . . . . 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 2324	. . . 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 2310	. . 3 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
44	24	nnzi 9505	. . . . 5 |- ( Base ` ndx ) e. ZZ
45	2	nnzd 9606	. . . . 5 |- ( ph -> I e. ZZ )
46		zdceq 9560	. . . . 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 843	. . . 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 805	. 2 |- ( ph -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
51	50, 21	prssd 3833	1 |- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2201   _Vcvv 2801   \ cdif 3196   u. cun 3197   i^i cin 3198   C_ wss 3199   {csn 3670   {cpr 3671   <.cop 3673   class class class wbr 4089   dom cdm 4727   |` cres 4729   ` cfv 5328  (class class class)co 6023   NNcn 9148   ZZcz 9484   Struct cstr 13101   ndxcnx 13102   sSet csts 13103   Basecbs 13105

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-n0 9408  df-z 9485  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112

This theorem is referenced by:  setsvtx  15931  setsiedg  15932