Theorem stgrusgra 47953

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 6802	. . . . 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 6763	. . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → 𝑘 ∈ 𝒫 (0...𝑁))
5		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑘 = {0, 𝑥} → (♯‘𝑘) = (♯‘{0, 𝑥}))
6		0red 11118	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 ∈ ℝ)
7		elfznn 13456	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
8	7	nngt0d 12177	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 < 𝑥)
9	6, 8	ltned 11252	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → 0 ≠ 𝑥)
10		c0ex 11109	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
11		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12	10, 11	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V ∧ 𝑥 ∈ V)
13		hashprg 14302	. . . . . . . . . . . . . 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 2786	. . . . . . . . . 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 3130	. . . . . . 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 2733	. . . . . . . 8 ⊢ (𝑒 = 𝑘 → (𝑒 = {0, 𝑥} ↔ 𝑘 = {0, 𝑥}))
23	22	rexbidv 3153	. . . . . . 7 ⊢ (𝑒 = 𝑘 → (∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}))
24	23	elrab 3648	. . . . . 6 ⊢ (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ↔ (𝑘 ∈ 𝒫 (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}))
25		fveqeq2 6831	. . . . . . 7 ⊢ (𝑒 = 𝑘 → ((♯‘𝑒) = 2 ↔ (♯‘𝑘) = 2))
26	25	elrab 3648	. . . . . 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 3941	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ⊆ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
29		f1ss 6725	. . . 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 47949	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
32	31	dmeqd 5848	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
33		dmresi 6003	. . . . 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 2780	. . . 4 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
35		stgrvtx 47948	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
36	35	pweqd 4568	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝒫 (Vtx‘(StarGr‘𝑁)) = 𝒫 (0...𝑁))
37	36	rabeqdv 3410	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
38	31, 34, 37	f1eq123d 6756	. . 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 6835	. . 3 ⊢ (StarGr‘𝑁) ∈ V
41		eqid 2729	. . . 4 ⊢ (Vtx‘(StarGr‘𝑁)) = (Vtx‘(StarGr‘𝑁))
42		eqid 2729	. . . 4 ⊢ (iEdg‘(StarGr‘𝑁)) = (iEdg‘(StarGr‘𝑁))
43	41, 42	isusgrs 29101	. . 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 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394  Vcvv 3436   ⊆ wss 3903  𝒫 cpw 4551  {cpr 4579   I cid 5513  dom cdm 5619   ↾ cres 5621  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  0cc0 11009  1c1 11010  2c2 12183  ℕ₀cn0 12384  ...cfz 13410  ♯chash 14237  Vtxcvtx 28941  iEdgciedg 28942  USGraphcusgr 29094  StarGrcstgr 47945

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-hash 14238  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-edgf 28934  df-vtx 28943  df-iedg 28944  df-usgr 29096  df-stgr 47946

This theorem is referenced by:  isubgr3stgrlem8  47967  isubgr3stgr  47969