Theorem stgrusgra 47926

Description: The star graph S_N is a simple graph. (Contributed by AV, 11-Sep-2025.)

Assertion

Ref	Expression
stgrusgra	⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)

Proof of Theorem stgrusgra

Dummy variables 𝑒 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6886	. . . . 5 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1-onto→{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}
2		f1of1 6847	. . . . 5 ⊢ (( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1-onto→{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} → ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
3	1, 2	mp1i 13	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
4		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → 𝑘 ∈ 𝒫 (0...𝑁))
5		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑘 = {0, 𝑥} → (♯‘𝑘) = (♯‘{0, 𝑥}))
6		0red 11264	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 ∈ ℝ)
7		elfznn 13593	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
8	7	nngt0d 12315	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 < 𝑥)
9	6, 8	ltned 11397	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → 0 ≠ 𝑥)
10		c0ex 11255	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
11		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12	10, 11	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V ∧ 𝑥 ∈ V)
13		hashprg 14434	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ V ∧ 𝑥 ∈ V) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
14	12, 13	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
15	9, 14	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑁) → (♯‘{0, 𝑥}) = 2)
16	15	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (♯‘{0, 𝑥}) = 2)
17	5, 16	sylan9eqr 2799	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (♯‘𝑘) = 2)
18	4, 17	jca 511	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2))
19	18	ex 412	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
20	19	rexlimdva 3155	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) → (∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
21	20	expimpd 453	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑘 ∈ 𝒫 (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
22		eqeq1 2741	. . . . . . . 8 ⊢ (𝑒 = 𝑘 → (𝑒 = {0, 𝑥} ↔ 𝑘 = {0, 𝑥}))
23	22	rexbidv 3179	. . . . . . 7 ⊢ (𝑒 = 𝑘 → (∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}))
24	23	elrab 3692	. . . . . 6 ⊢ (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ↔ (𝑘 ∈ 𝒫 (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}))
25		fveqeq2 6915	. . . . . . 7 ⊢ (𝑒 = 𝑘 → ((♯‘𝑒) = 2 ↔ (♯‘𝑘) = 2))
26	25	elrab 3692	. . . . . 6 ⊢ (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2} ↔ (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2))
27	21, 24, 26	3imtr4g 296	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} → 𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2}))
28	27	ssrdv 3989	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ⊆ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
29		f1ss 6809	. . . 4 ⊢ ((( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ∧ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ⊆ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2}) → ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
30	3, 28, 29	syl2anc 584	. . 3 ⊢ (𝑁 ∈ ℕ0 → ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
31		stgriedg 47922	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
32	31	dmeqd 5916	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
33		dmresi 6070	. . . . 5 ⊢ dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}
34	32, 33	eqtrdi 2793	. . . 4 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
35		stgrvtx 47921	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
36	35	pweqd 4617	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝒫 (Vtx‘(StarGr‘𝑁)) = 𝒫 (0...𝑁))
37	36	rabeqdv 3452	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
38	31, 34, 37	f1eq123d 6840	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} ↔ ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}–1-1→{𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2}))
39	30, 38	mpbird 257	. 2 ⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2})
40		fvex 6919	. . 3 ⊢ (StarGr‘𝑁) ∈ V
41		eqid 2737	. . . 4 ⊢ (Vtx‘(StarGr‘𝑁)) = (Vtx‘(StarGr‘𝑁))
42		eqid 2737	. . . 4 ⊢ (iEdg‘(StarGr‘𝑁)) = (iEdg‘(StarGr‘𝑁))
43	41, 42	isusgrs 29173	. . 3 ⊢ ((StarGr‘𝑁) ∈ V → ((StarGr‘𝑁) ∈ USGraph ↔ (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2}))
44	40, 43	mp1i 13	. 2 ⊢ (𝑁 ∈ ℕ0 → ((StarGr‘𝑁) ∈ USGraph ↔ (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2}))
45	39, 44	mpbird 257	1 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  {cpr 4628   I cid 5577  dom cdm 5685   ↾ cres 5687  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  2c2 12321  ℕ₀cn0 12526  ...cfz 13547  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  USGraphcusgr 29166  StarGrcstgr 47918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-hash 14370  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-edgf 29004  df-vtx 29015  df-iedg 29016  df-usgr 29168  df-stgr 47919

This theorem is referenced by:  isubgr3stgrlem8  47940  isubgr3stgr  47942