Theorem isubgr3stgrlem3 49269

Description: Lemma 3 for isubgr3stgr 49276. (Contributed by AV, 17-Sep-2025.)

Hypotheses

Ref	Expression
isubgr3stgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c	⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.n	⊢ 𝑁 ∈ ℕ0
isubgr3stgr.s	⊢ 𝑆 = (StarGr‘𝑁)
isubgr3stgr.w	⊢ 𝑊 = (Vtx‘𝑆)

Assertion

Ref	Expression
isubgr3stgrlem3	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0))

Distinct variable groups:   𝐶,𝑔   𝑔,𝑊   𝑔,𝑋

Allowed substitution hints:   𝑆(𝑔)   𝑈(𝑔)   𝐺(𝑔)   𝑁(𝑔)   𝑉(𝑔)

Proof of Theorem isubgr3stgrlem3

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgr3stgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		isubgr3stgr.u	. . 3 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
3		isubgr3stgr.c	. . 3 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
4		isubgr3stgr.n	. . 3 ⊢ 𝑁 ∈ ℕ0
5		isubgr3stgr.s	. . 3 ⊢ 𝑆 = (StarGr‘𝑁)
6		isubgr3stgr.w	. . 3 ⊢ 𝑊 = (Vtx‘𝑆)
7	1, 2, 3, 4, 5, 6	isubgr3stgrlem2 49268	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}))
8		f1odm 6847	. . . 4 ⊢ (𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) → dom 𝑓 = 𝑈)
9		simpr 490	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}))
10		simpl2 1211	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → 𝑋 ∈ 𝑉)
11		c0ex 11324	. . . . . . . 8 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → 0 ∈ V)
13		neldifsnd 4756	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → ¬ 0 ∈ (𝑊 ∖ {0}))
14		df-nel 3066	. . . . . . . 8 ⊢ (0 ∉ (𝑊 ∖ {0}) ↔ ¬ 0 ∈ (𝑊 ∖ {0}))
15	13, 14	sylibr 237	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → 0 ∉ (𝑊 ∖ {0}))
16		eqid 2763	. . . . . . . 8 ⊢ (𝑓 ∪ {⟨𝑋, 0⟩}) = (𝑓 ∪ {⟨𝑋, 0⟩})
17	1, 2, 3, 16	isubgr3stgrlem1 49267	. . . . . . 7 ⊢ ((𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) ∧ 𝑋 ∈ 𝑉 ∧ (0 ∈ V ∧ 0 ∉ (𝑊 ∖ {0}))) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}))
18	9, 10, 12, 15, 17	syl112anc 1402	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0})) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}))
19	18	ex 418	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0})))
20		f1of 6843	. . . . . . . . 9 ⊢ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
21	20	3ad2ant2 1152	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
22	3	ovexi 7474	. . . . . . . . 9 ⊢ 𝐶 ∈ V
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 𝐶 ∈ V)
24	21, 23	fexd 7253	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}) ∈ V)
25	5, 6	stgrvtx0 49263	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊)
26	4, 25	mp1i 14	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → 0 ∈ 𝑊)
27	26	snssd 4747	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → {0} ⊆ 𝑊)
28		undifr 4442	. . . . . . . . . . . 12 ⊢ ({0} ⊆ 𝑊 ↔ ((𝑊 ∖ {0}) ∪ {0}) = 𝑊)
29	27, 28	sylib 221	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑊 ∖ {0}) ∪ {0}) = 𝑊)
30	29	f1oeq3d 6840	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊))
31	30	biimpa 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0})) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊)
32	31	3adant3 1150	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊)
33		simp12 1223	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 𝑋 ∈ 𝑉)
34	11	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 0 ∈ V)
35		nbgrnself2 29968	. . . . . . . . . . 11 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
36		df-nel 3066	. . . . . . . . . . . 12 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
37	2	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
38	36, 37	xchbinxr 338	. . . . . . . . . . 11 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ 𝑈)
39	35, 38	mpbi 233	. . . . . . . . . 10 ⊢ ¬ 𝑋 ∈ 𝑈
40		eleq2 2853	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑈 → (𝑋 ∈ dom 𝑓 ↔ 𝑋 ∈ 𝑈))
41	40	notbid 321	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑈 → (¬ 𝑋 ∈ dom 𝑓 ↔ ¬ 𝑋 ∈ 𝑈))
42	41	3ad2ant3 1153	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (¬ 𝑋 ∈ dom 𝑓 ↔ ¬ 𝑋 ∈ 𝑈))
43	39, 42	mpbiri 261	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ¬ 𝑋 ∈ dom 𝑓)
44		fsnunfv 7212	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 0 ∈ V ∧ ¬ 𝑋 ∈ dom 𝑓) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
45	33, 34, 43, 44	syl3anc 1399	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
46	32, 45	jca 521	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊 ∧ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
47		f1oeq1 6831	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔:𝐶⟶1-1onto𝑊 ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊))
48		fveq1 6904	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔‘𝑋) = ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋))
49	48	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔‘𝑋) = 0 ↔ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
50	47, 49	anbi12d 644	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0) ↔ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto𝑊 ∧ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)))
51	24, 46, 50	spcedv 3556	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0))
52	51	3exp 1137	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶1-1onto((𝑊 ∖ {0}) ∪ {0}) → (dom 𝑓 = 𝑈 → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0))))
53	19, 52	syld 48	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) → (dom 𝑓 = 𝑈 → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0))))
54	8, 53	mpdi 46	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0)))
55	54	exlimdv 1968	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (∃𝑓 𝑓:𝑈⟶1-1onto(𝑊 ∖ {0}) → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0)))
56	7, 55	mpd 16	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶⟶1-1onto𝑊 ∧ (𝑔‘𝑋) = 0))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ∉ wnel 3065  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  {csn 4588  ⟨cop 4838  dom cdm 5655  ⟶wf 6551  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  0cc0 11224  ℕ₀cn0 12630  ♯chash 14498  Vtxcvtx 29601  USGraphcusgr 29757   NeighbVtx cnbgr 29940   ClNeighbVtx cclnbgr 49119  StarGrcstgr 49252

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-xnn0 12704  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-hash 14499  df-struct 17351  df-slot 17386  df-ndx 17398  df-base 17414  df-edgf 29594  df-vtx 29603  df-nbgr 29941  df-clnbgr 49120  df-stgr 49253

This theorem is used by:  isubgr3stgr  49276