Theorem bassetsnn 13356

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 3723	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ {𝐼})
4	2, 3	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
5		basprssdmsets.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
6		dmsnopg 5239	. . . . . . . . . 10 ⊢ (𝐸 ∈ 𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
7	5, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
8	4, 7	eleqtrrd 2314	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ dom {⟨𝐼, 𝐸⟩})
9		elun2 3391	. . . . . . . 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 4968	. . . . . . 7 ⊢ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
12	10, 11	eleqtrrdi 2328	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
13		basprssdmsets.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 Struct 𝑋)
14		structex 13311	. . . . . . . . 9 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
15	13, 14	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ V)
16		opexg 4349	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑊) → ⟨𝐼, 𝐸⟩ ∈ V)
17	2, 5, 16	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐼, 𝐸⟩ ∈ V)
18		setsvalg 13329	. . . . . . . 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 4963	. . . . . 6 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
21	12, 20	eleqtrrd 2314	. . . . 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 2311	. . 3 ⊢ ((𝜑 ∧ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
24		basendxnn 13355	. . . . . . . . . . 11 ⊢ (Base‘ndx) ∈ ℕ
25	24	elexi 2828	. . . . . . . . . 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 2313	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → (Base‘ndx) ∈ {𝐼})
30		elsni 3712	. . . . . . . . . . 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 1472	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → ¬ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩})
33	26, 32	eldifd 3224	. . . . . . . 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 3408	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆))
37		dmres 5064	. . . . . . 7 ⊢ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆)
38	36, 37	eleqtrrdi 2328	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})))
39		elun1 3390	. . . . . 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 2328	. . . 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 2314	. . 3 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
44	24	nnzi 9618	. . . . 5 ⊢ (Base‘ndx) ∈ ℤ
45	2	nnzd 9720	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ)
46		zdceq 9673	. . . . 5 ⊢ (((Base‘ndx) ∈ ℤ ∧ 𝐼 ∈ ℤ) → DECID (Base‘ndx) = 𝐼)
47	44, 45, 46	sylancr 414	. . . 4 ⊢ (𝜑 → DECID (Base‘ndx) = 𝐼)
48		exmiddc 844	. . . 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 806	. 2 ⊢ (𝜑 → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
51	50, 21	prssd 3858	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∖ cdif 3211   ∪ cun 3212   ∩ cin 3213   ⊆ wss 3214  {csn 3694  {cpr 3695  ⟨cop 3697   class class class wbr 4114  dom cdm 4754   ↾ cres 4756  ‘cfv 5357  (class class class)co 6058  ℕcn 9257  ℤcz 9597   Struct cstr 13295  ndxcnx 13296   sSet csts 13297  Basecbs 13299

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-inn 9258  df-n0 9517  df-z 9598  df-struct 13301  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306

This theorem is referenced by:  setsvtx  16175  setsiedg  16176