Theorem bassetsnn 13055

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 3675	. . . . . . . . . 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 5176	. . . . . . . . . 10 |- ( E e. W -> dom { <. I , E >. } = { I } )
7	5, 6	syl 14	. . . . . . . . 9 |- ( ph -> dom { <. I , E >. } = { I } )
8	4, 7	eleqtrrd 2289	. . . . . . . 8 |- ( ph -> I e. dom { <. I , E >. } )
9		elun2 3352	. . . . . . . 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 4907	. . . . . . 7 |- dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) = ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } )
12	10, 11	eleqtrrdi 2303	. . . . . 6 |- ( ph -> I e. dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
13		basprssdmsets.s	. . . . . . . . 9 |- ( ph -> S Struct X )
14		structex 13010	. . . . . . . . 9 |- ( S Struct X -> S e. _V )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> S e. _V )
16		opexg 4293	. . . . . . . . 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 13028	. . . . . . . 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 4902	. . . . . 6 |- ( ph -> dom ( S sSet <. I , E >. ) = dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
21	12, 20	eleqtrrd 2289	. . . . 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 2286	. . 3 |- ( ( ph /\ ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
24		basendxnn 13054	. . . . . . . . . . 11 |- ( Base ` ndx ) e. NN
25	24	elexi 2792	. . . . . . . . . 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 2288	. . . . . . . . . . 11 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) e. { I } )
30		elsni 3664	. . . . . . . . . . 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 1449	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> -. ( Base ` ndx ) e. dom { <. I , E >. } )
33	26, 32	eldifd 3187	. . . . . . . 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 3369	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( ( _V \ dom { <. I , E >. } ) i^i dom S ) )
37		dmres 5002	. . . . . . 7 |- dom ( S |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom S )
38	36, 37	eleqtrrdi 2303	. . . . . 6 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S |` ( _V \ dom { <. I , E >. } ) ) )
39		elun1 3351	. . . . . 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 2303	. . . 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 2289	. . 3 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
44	24	nnzi 9435	. . . . 5 |- ( Base ` ndx ) e. ZZ
45	2	nnzd 9536	. . . . 5 |- ( ph -> I e. ZZ )
46		zdceq 9490	. . . . 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 840	. . . 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 802	. 2 |- ( ph -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
51	50, 21	prssd 3806	1 |- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 712  DECID wdc 838   = wceq 1375   e. wcel 2180   _Vcvv 2779   \ cdif 3174   u. cun 3175   i^i cin 3176   C_ wss 3177   {csn 3646   {cpr 3647   <.cop 3649   class class class wbr 4062   dom cdm 4696   |` cres 4698   ` cfv 5294  (class class class)co 5974   NNcn 9078   ZZcz 9414   Struct cstr 12994   ndxcnx 12995   sSet csts 12996   Basecbs 12998

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-n0 9338  df-z 9415  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005

This theorem is referenced by:  setsvtx  15817  setsiedg  15818