Theorem bassetsnn 13138

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	⊢ (𝜑 → 𝑆 Struct 𝑋)
bassetsnn.i	⊢ (𝜑 → 𝐼 ∈ ℕ)
basprssdmsets.w	⊢ (𝜑 → 𝐸 ∈ 𝑊)
basprssdmsets.b	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)

Assertion

Ref	Expression
bassetsnn	⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Proof of Theorem bassetsnn

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ (Base‘ndx) = 𝐼) → (Base‘ndx) = 𝐼)
2		bassetsnn.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
3		snidg 3698	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ {𝐼})
4	2, 3	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
5		basprssdmsets.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
6		dmsnopg 5208	. . . . . . . . . 10 ⊢ (𝐸 ∈ 𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
7	5, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
8	4, 7	eleqtrrd 2311	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ dom {⟨𝐼, 𝐸⟩})
9		elun2 3375	. . . . . . . 8 ⊢ (𝐼 ∈ dom {⟨𝐼, 𝐸⟩} → 𝐼 ∈ (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}))
10	8, 9	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}))
11		dmun 4938	. . . . . . 7 ⊢ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
12	10, 11	eleqtrrdi 2325	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
13		basprssdmsets.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 Struct 𝑋)
14		structex 13093	. . . . . . . . 9 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
15	13, 14	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ V)
16		opexg 4320	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑊) → ⟨𝐼, 𝐸⟩ ∈ V)
17	2, 5, 16	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐼, 𝐸⟩ ∈ V)
18		setsvalg 13111	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ ⟨𝐼, 𝐸⟩ ∈ V) → (𝑆 sSet ⟨𝐼, 𝐸⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
19	15, 17, 18	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝑆 sSet ⟨𝐼, 𝐸⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
20	19	dmeqd 4933	. . . . . 6 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
21	12, 20	eleqtrrd 2311	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
22	21	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (Base‘ndx) = 𝐼) → 𝐼 ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
23	1, 22	eqeltrd 2308	. . 3 ⊢ ((𝜑 ∧ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
24		basendxnn 13137	. . . . . . . . . . 11 ⊢ (Base‘ndx) ∈ ℕ
25	24	elexi 2815	. . . . . . . . . 10 ⊢ (Base‘ndx) ∈ V
26	25	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ V)
27		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩})
28	7	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → dom {⟨𝐼, 𝐸⟩} = {𝐼})
29	27, 28	eleqtrd 2310	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → (Base‘ndx) ∈ {𝐼})
30		elsni 3687	. . . . . . . . . . 11 ⊢ ((Base‘ndx) ∈ {𝐼} → (Base‘ndx) = 𝐼)
31	29, 30	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → (Base‘ndx) = 𝐼)
32	31	stoic1a 1471	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → ¬ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩})
33	26, 32	eldifd 3210	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ (V ∖ dom {⟨𝐼, 𝐸⟩}))
34		basprssdmsets.b	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
35	34	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom 𝑆)
36	33, 35	elind 3392	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆))
37		dmres 5034	. . . . . . 7 ⊢ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆)
38	36, 37	eleqtrrdi 2325	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})))
39		elun1 3374	. . . . . 6 ⊢ ((Base‘ndx) ∈ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) → (Base‘ndx) ∈ (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}))
40	38, 39	syl 14	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}))
41	40, 11	eleqtrrdi 2325	. . . 4 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
42	20	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
43	41, 42	eleqtrrd 2311	. . 3 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
44	24	nnzi 9499	. . . . 5 ⊢ (Base‘ndx) ∈ ℤ
45	2	nnzd 9600	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ)
46		zdceq 9554	. . . . 5 ⊢ (((Base‘ndx) ∈ ℤ ∧ 𝐼 ∈ ℤ) → DECID (Base‘ndx) = 𝐼)
47	44, 45, 46	sylancr 414	. . . 4 ⊢ (𝜑 → DECID (Base‘ndx) = 𝐼)
48		exmiddc 843	. . . 4 ⊢ (DECID (Base‘ndx) = 𝐼 → ((Base‘ndx) = 𝐼 ∨ ¬ (Base‘ndx) = 𝐼))
49	47, 48	syl 14	. . 3 ⊢ (𝜑 → ((Base‘ndx) = 𝐼 ∨ ¬ (Base‘ndx) = 𝐼))
50	23, 43, 49	mpjaodan 805	. 2 ⊢ (𝜑 → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
51	50, 21	prssd 3832	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  {csn 3669  {cpr 3670  ⟨cop 3672   class class class wbr 4088  dom cdm 4725   ↾ cres 4727  ‘cfv 5326  (class class class)co 6017  ℕcn 9142  ℤcz 9478   Struct cstr 13077  ndxcnx 13078   sSet csts 13079  Basecbs 13081

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-struct 13083  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088

This theorem is referenced by:  setsvtx  15901  setsiedg  15902