Theorem isubgr3stgrlem3 48657

Description: Lemma 3 for isubgr3stgr 48664. (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 48656	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
8		f1odm 6825	. . . 4 ⊢ (𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}) → dom 𝑓 = 𝑈)
9		simpr 489	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))
10		simpl2 1209	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 𝑋 ∈ 𝑉)
11		c0ex 11200	. . . . . . . 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 4763	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → ¬ 0 ∈ (𝑊 ∖ {0}))
14		df-nel 3071	. . . . . . . 8 ⊢ (0 ∉ (𝑊 ∖ {0}) ↔ ¬ 0 ∈ (𝑊 ∖ {0}))
15	13, 14	sylibr 237	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) ∧ 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) → 0 ∉ (𝑊 ∖ {0}))
16		eqid 2769	. . . . . . . 8 ⊢ (𝑓 ∪ {⟨𝑋, 0⟩}) = (𝑓 ∪ {⟨𝑋, 0⟩})
17	1, 2, 3, 16	isubgr3stgrlem1 48655	. . . . . . 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 1399	. . . . . 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 417	. . . . 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 6821	. . . . . . . . 9 ⊢ ((𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→((𝑊 ∖ {0}) ∪ {0}) → (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶⟶((𝑊 ∖ {0}) ∪ {0}))
21	20	3ad2ant2 1150	. . . . . . . 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 7445	. . . . . . . . 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 7226	. . . . . . 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 48651	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊)
26	4, 25	mp1i 14	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → 0 ∈ 𝑊)
27	26	snssd 4755	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → {0} ⊆ 𝑊)
28		undifr 4447	. . . . . . . . . . . 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 6818	. . . . . . . . . 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 481	. . . . . . . . 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 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–1-1-onto→𝑊)
33		simp12 1221	. . . . . . . . 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 29651	. . . . . . . . . . 11 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
36		df-nel 3071	. . . . . . . . . . . 12 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
37	2	eleq2i 2861	. . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 1151	. . . . . . . . . 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 261	. . . . . . . . 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 7186	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 0 ∈ V ∧ ¬ 𝑋 ∈ dom 𝑓) → ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0)
45	33, 34, 43, 44	syl3anc 1396	. . . . . . . 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 520	. . . . . . 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 6809	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔:𝐶–1-1-onto→𝑊 ↔ (𝑓 ∪ {⟨𝑋, 0⟩}):𝐶–1-1-onto→𝑊))
48		fveq1 6881	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → (𝑔‘𝑋) = ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋))
49	48	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑋, 0⟩}) → ((𝑔‘𝑋) = 0 ↔ ((𝑓 ∪ {⟨𝑋, 0⟩})‘𝑋) = 0))
50	47, 49	anbi12d 643	. . . . . . 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 3564	. . . . . 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 1135	. . . . 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 48	. . . 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 46	. . 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 1960	. 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 16	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶–1-1-onto→𝑊 ∧ (𝑔‘𝑋) = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ∉ wnel 3070  Vcvv 3461   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  {csn 4592  ⟨cop 4598  dom cdm 5662  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  0cc0 11100  ℕ₀cn0 12504  ♯chash 14366  Vtxcvtx 29287  USGraphcusgr 29440   NeighbVtx cnbgr 29623   ClNeighbVtx cclnbgr 48507  StarGrcstgr 48640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-xnn0 12578  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-hash 14367  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-edgf 29280  df-vtx 29289  df-nbgr 29624  df-clnbgr 48508  df-stgr 48641

This theorem is referenced by:  isubgr3stgr  48664