Theorem bassetsnn 13461

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 3738	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ {𝐼})
4	2, 3	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
5		basprssdmsets.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
6		dmsnopg 5259	. . . . . . . . . 10 ⊢ (𝐸 ∈ 𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
7	5, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
8	4, 7	eleqtrrd 2318	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ dom {⟨𝐼, 𝐸⟩})
9		elun2 3397	. . . . . . . 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 4988	. . . . . . 7 ⊢ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
12	10, 11	eleqtrrdi 2332	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
13		basprssdmsets.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 Struct 𝑋)
14		structex 13416	. . . . . . . . 9 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
15	13, 14	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ V)
16		opexg 4368	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑊) → ⟨𝐼, 𝐸⟩ ∈ V)
17	2, 5, 16	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐼, 𝐸⟩ ∈ V)
18		setsvalg 13434	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ ⟨𝐼, 𝐸⟩ ∈ V) → (𝑆 sSet ⟨𝐼, 𝐸⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
19	15, 17, 18	syl2anc 415	. . . . . . 7 ⊢ (𝜑 → (𝑆 sSet ⟨𝐼, 𝐸⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
20	19	dmeqd 4983	. . . . . 6 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = dom ((𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
21	12, 20	eleqtrrd 2318	. . . . 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 2315	. . 3 ⊢ ((𝜑 ∧ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
24		basendxnn 13460	. . . . . . . . . . 11 ⊢ (Base‘ndx) ∈ ℕ
25	24	elexi 2834	. . . . . . . . . 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 2317	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩}) → (Base‘ndx) ∈ {𝐼})
30		elsni 3727	. . . . . . . . . . 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 1476	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → ¬ (Base‘ndx) ∈ dom {⟨𝐼, 𝐸⟩})
33	26, 32	eldifd 3230	. . . . . . . 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 3414	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆))
37		dmres 5084	. . . . . . 7 ⊢ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝑆)
38	36, 37	eleqtrrdi 2332	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})))
39		elun1 3396	. . . . . 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 2332	. . . 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 2318	. . 3 ⊢ ((𝜑 ∧ ¬ (Base‘ndx) = 𝐼) → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
44	24	nnzi 9670	. . . . 5 ⊢ (Base‘ndx) ∈ ℤ
45	2	nnzd 9772	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ)
46		zdceq 9725	. . . . 5 ⊢ (((Base‘ndx) ∈ ℤ ∧ 𝐼 ∈ ℤ) → DECID (Base‘ndx) = 𝐼)
47	44, 45, 46	sylancr 418	. . . 4 ⊢ (𝜑 → DECID (Base‘ndx) = 𝐼)
48		exmiddc 848	. . . 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 810	. 2 ⊢ (𝜑 → (Base‘ndx) ∈ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
51	50, 21	prssd 3874	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  {csn 3709  {cpr 3710  ⟨cop 3712   class class class wbr 4130  dom cdm 4774   ↾ cres 4776  ‘cfv 5377  (class class class)co 6085  ℕcn 9307  ℤcz 9649   Struct cstr 13400  ndxcnx 13401   sSet csts 13402  Basecbs 13404

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-struct 13406  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411

This theorem is used by:  setsvtx  16463  setsiedg  16464