Theorem isubgr3stgrlem3 48473

Description: Lemma 3 for isubgr3stgr 48480. (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 48472	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
8		f1odm 6775	. . . 4 ⊢ (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → dom 𝑓 = 𝑈)
9		simpr 486	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
10		simpl2 1200	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑋 ∈ 𝑉)
11		c0ex 11133	. . . . . . . 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 4729	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → ¬ 0 ∈ (𝑊 ∖ {0}))
14		df-nel 3041	. . . . . . . 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 2741	. . . . . . . 8 ⊢ (𝑓 ∪ {⟨𝑋, 0⟩}) = (𝑓 ∪ {⟨𝑋, 0⟩})
17	1, 2, 3, 16	isubgr3stgrlem1 48471	. . . . . . 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 1383	. . . . . 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 414	. . . . 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 6771	. . . . . . . . 9 ⊢ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
21	20	3ad2ant2 1141	. . . . . . . 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 7394	. . . . . . . . 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 7175	. . . . . . 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 48467	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊)
26	4, 25	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → 0 ∈ 𝑊)
27	26	snssd 4721	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → {0} ⊆ 𝑊)
28		undifr 4414	. . . . . . . . . . . 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 6768	. . . . . . . . . 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 478	. . . . . . . . 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 1139	. . . . . . . 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 1212	. . . . . . . . 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 29451	. . . . . . . . . . 11 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
36		df-nel 3041	. . . . . . . . . . . 12 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
37	2	eleq2i 2833	. . . . . . . . . . . 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 2830	. . . . . . . . . . . 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 1142	. . . . . . . . . 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 7135	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 0 ∈ V ∧ ¬ 𝑋 ∈ dom 𝑓) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
45	33, 34, 43, 44	syl3anc 1380	. . . . . . . 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 517	. . . . . . 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 6759	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔:𝐶–1-1-onto→𝑊 ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊))
48		fveq1 6830	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔‘𝑋) = ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋))
49	48	eqeq1d 2743	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔‘𝑋) = 0 ↔ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
50	47, 49	anbi12d 639	. . . . . . 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 3538	. . . . . 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 1126	. . . . 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 1941	. 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 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ∉ wnel 3040  Vcvv 3433   ∖ cdif 3882   ∪ cun 3883   ⊆ wss 3885  {csn 4558  ⟨cop 4564  dom cdm 5621  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  0cc0 11033  ℕ₀cn0 12432  ♯chash 14287  Vtxcvtx 29087  USGraphcusgr 29240   NeighbVtx cnbgr 29423   ClNeighbVtx cclnbgr 48323  StarGrcstgr 48456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-xnn0 12506  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-hash 14288  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-edgf 29080  df-vtx 29089  df-nbgr 29424  df-clnbgr 48324  df-stgr 48457

This theorem is referenced by:  isubgr3stgr  48480