Theorem isubgr3stgrlem3 48595

Description: Lemma 3 for isubgr3stgr 48602. (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–1-1-onto→𝑊 ∧ (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 48594	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
8		f1odm 6812	. . . 4 ⊢ (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → dom 𝑓 = 𝑈)
9		simpr 488	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
10		simpl2 1207	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑋 ∈ 𝑉)
11		c0ex 11175	. . . . . . . 8 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 0 ∈ V)
13		neldifsnd 4755	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → ¬ 0 ∈ (𝑊 ∖ {0}))
14		df-nel 3064	. . . . . . . 8 ⊢ (0 ∉ (𝑊 ∖ {0}) ↔ ¬ 0 ∈ (𝑊 ∖ {0}))
15	13, 14	sylibr 236	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 0 ∉ (𝑊 ∖ {0}))
16		eqid 2764	. . . . . . . 8 ⊢ (𝑓 ∪ {⟨𝑋, 0⟩}) = (𝑓 ∪ {⟨𝑋, 0⟩})
17	1, 2, 3, 16	isubgr3stgrlem1 48593	. . . . . . 7 ⊢ ((𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) ∧ 𝑋 ∈ 𝑉 ∧ (0 ∈ V ∧ 0 ∉ (𝑊 ∖ {0}))) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}))
18	9, 10, 12, 15, 17	syl112anc 1395	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}))
19	18	ex 416	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0})))
20		f1of 6808	. . . . . . . . 9 ⊢ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
21	20	3ad2ant2 1148	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
22	3	ovexi 7432	. . . . . . . . 9 ⊢ 𝐶 ∈ V
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 𝐶 ∈ V)
24	21, 23	fexd 7213	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}) ∈ V)
25	5, 6	stgrvtx0 48589	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊)
26	4, 25	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → 0 ∈ 𝑊)
27	26	snssd 4747	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → {0} ⊆ 𝑊)
28		undifr 4439	. . . . . . . . . . . 12 ⊢ ({0} ⊆ 𝑊 ↔ ((𝑊 ∖ {0}) ∪ {0}) = 𝑊)
29	27, 28	sylib 220	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑊 ∖ {0}) ∪ {0}) = 𝑊)
30	29	f1oeq3d 6805	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊))
31	30	biimpa 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0})) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊)
32	31	3adant3 1146	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊)
33		simp12 1219	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 𝑋 ∈ 𝑉)
34	11	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → 0 ∈ V)
35		nbgrnself2 29563	. . . . . . . . . . 11 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
36		df-nel 3064	. . . . . . . . . . . 12 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
37	2	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
38	36, 37	xchbinxr 337	. . . . . . . . . . 11 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ 𝑈)
39	35, 38	mpbi 232	. . . . . . . . . 10 ⊢ ¬ 𝑋 ∈ 𝑈
40		eleq2 2853	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑈 → (𝑋 ∈ dom 𝑓 ↔ 𝑋 ∈ 𝑈))
41	40	notbid 320	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑈 → (¬ 𝑋 ∈ dom 𝑓 ↔ ¬ 𝑋 ∈ 𝑈))
42	41	3ad2ant3 1149	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → (¬ 𝑋 ∈ dom 𝑓 ↔ ¬ 𝑋 ∈ 𝑈))
43	39, 42	mpbiri 260	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ¬ 𝑋 ∈ dom 𝑓)
44		fsnunfv 7173	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 0 ∈ V ∧ ¬ 𝑋 ∈ dom 𝑓) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
45	33, 34, 43, 44	syl3anc 1392	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
46	32, 45	jca 519	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊 ∧ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
47		f1oeq1 6796	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔:𝐶–1-1-onto→𝑊 ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊))
48		fveq1 6868	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔‘𝑋) = ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋))
49	48	eqeq1d 2766	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔‘𝑋) = 0 ↔ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
50	47, 49	anbi12d 641	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0) ↔ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊 ∧ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)))
51	24, 46, 50	spcedv 3559	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) ∧ dom 𝑓 = 𝑈) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))
52	51	3exp 1133	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) → (dom 𝑓 = 𝑈 → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))))
53	19, 52	syld 47	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → (dom 𝑓 = 𝑈 → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))))
54	8, 53	mpdi 45	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0)))
55	54	exlimdv 1955	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0)))
56	7, 55	mpd 15	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ∉ wnel 3063  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  {csn 4584  ⟨cop 4590  dom cdm 5649  ⟶wf 6519  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  0cc0 11075  ℕ₀cn0 12483  ♯chash 14345  Vtxcvtx 29199  USGraphcusgr 29352   NeighbVtx cnbgr 29535   ClNeighbVtx cclnbgr 48445  StarGrcstgr 48578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-xnn0 12557  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-hash 14346  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-edgf 29192  df-vtx 29201  df-nbgr 29536  df-clnbgr 48446  df-stgr 48579

This theorem is referenced by:  isubgr3stgr  48602