Theorem bassetsnn 13141

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 3698	. . . . . . . . . 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 5208	. . . . . . . . . 10 |- ( E e. W -> dom { <. I , E >. } = { I } )
7	5, 6	syl 14	. . . . . . . . 9 |- ( ph -> dom { <. I , E >. } = { I } )
8	4, 7	eleqtrrd 2311	. . . . . . . 8 |- ( ph -> I e. dom { <. I , E >. } )
9		elun2 3375	. . . . . . . 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 4938	. . . . . . 7 |- dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) = ( dom ( S |` ( _V \ dom { <. I , E >. } ) ) u. dom { <. I , E >. } )
12	10, 11	eleqtrrdi 2325	. . . . . 6 |- ( ph -> I e. dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
13		basprssdmsets.s	. . . . . . . . 9 |- ( ph -> S Struct X )
14		structex 13096	. . . . . . . . 9 |- ( S Struct X -> S e. _V )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> S e. _V )
16		opexg 4320	. . . . . . . . 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 13114	. . . . . . . 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 4933	. . . . . 6 |- ( ph -> dom ( S sSet <. I , E >. ) = dom ( ( S |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
21	12, 20	eleqtrrd 2311	. . . . 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 2308	. . 3 |- ( ( ph /\ ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
24		basendxnn 13140	. . . . . . . . . . 11 |- ( Base ` ndx ) e. NN
25	24	elexi 2815	. . . . . . . . . 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 2310	. . . . . . . . . . 11 |- ( ( ph /\ ( Base ` ndx ) e. dom { <. I , E >. } ) -> ( Base ` ndx ) e. { I } )
30		elsni 3687	. . . . . . . . . . 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 3210	. . . . . . . 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 3392	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. ( ( _V \ dom { <. I , E >. } ) i^i dom S ) )
37		dmres 5034	. . . . . . 7 |- dom ( S |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom S )
38	36, 37	eleqtrrdi 2325	. . . . . 6 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S |` ( _V \ dom { <. I , E >. } ) ) )
39		elun1 3374	. . . . . 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 2325	. . . 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 2311	. . 3 |- ( ( ph /\ -. ( Base ` ndx ) = I ) -> ( Base ` ndx ) e. dom ( S sSet <. I , E >. ) )
44	24	nnzi 9500	. . . . 5 |- ( Base ` ndx ) e. ZZ
45	2	nnzd 9601	. . . . 5 |- ( ph -> I e. ZZ )
46		zdceq 9555	. . . . 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 3832	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 2202   _Vcvv 2802   \ cdif 3197   u. cun 3198   i^i cin 3199   C_ wss 3200   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   dom cdm 4725   |` cres 4727   ` cfv 5326  (class class class)co 6018   NNcn 9143   ZZcz 9479   Struct cstr 13080   ndxcnx 13081   sSet csts 13082   Basecbs 13084

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-struct 13086  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091

This theorem is referenced by:  setsvtx  15905  setsiedg  15906