Theorem stgrusgra 48860

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 6860	. . . . 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 6820	. . . . 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 14	. . . 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 788	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → 𝑘 ∈ 𝒫 (0...𝑁))
5		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑘 = {0, 𝑥} → (♯‘𝑘) = (♯‘{0, 𝑥}))
6		0red 11236	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 ∈ ℝ)
7		elfznn 13608	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
8	7	nngt0d 12310	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 < 𝑥)
9	6, 8	ltned 11371	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → 0 ≠ 𝑥)
10		c0ex 11225	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
11		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12	10, 11	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V ∧ 𝑥 ∈ V)
13		hashprg 14459	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ V ∧ 𝑥 ∈ V) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
14	12, 13	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
15	9, 14	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑁) → (♯‘{0, 𝑥}) = 2)
16	15	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (♯‘{0, 𝑥}) = 2)
17	5, 16	sylan9eqr 2819	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (♯‘𝑘) = 2)
18	4, 17	jca 521	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2))
19	18	ex 418	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
20	19	rexlimdva 3165	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ 𝒫 (0...𝑁)) → (∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
21	20	expimpd 459	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑘 ∈ 𝒫 (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
22		eqeq1 2766	. . . . . . . 8 ⊢ (𝑒 = 𝑘 → (𝑒 = {0, 𝑥} ↔ 𝑘 = {0, 𝑥}))
23	22	rexbidv 3188	. . . . . . 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 6891	. . . . . . 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 299	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} → 𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2}))
28	27	ssrdv 3940	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ⊆ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
29		f1ss 6782	. . . 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 596	. . 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 48856	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
32	31	dmeqd 5893	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
33		dmresi 6052	. . . . 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 2813	. . . 4 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
35		stgrvtx 48855	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
36	35	pweqd 4577	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝒫 (Vtx‘(StarGr‘𝑁)) = 𝒫 (0...𝑁))
37	36	rabeqdv 3429	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
38	31, 34, 37	f1eq123d 6813	. . 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 260	. 2 ⊢ (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2})
40		fvex 6895	. . 3 ⊢ (StarGr‘𝑁) ∈ V
41		eqid 2762	. . . 4 ⊢ (Vtx‘(StarGr‘𝑁)) = (Vtx‘(StarGr‘𝑁))
42		eqid 2762	. . . 4 ⊢ (iEdg‘(StarGr‘𝑁)) = (iEdg‘(StarGr‘𝑁))
43	41, 42	isusgrs 29600	. . 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 14	. 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 260	1 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  {cpr 4589   I cid 5553  dom cdm 5659   ↾ cres 5661  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  0cc0 11125  1c1 11126  2c2 12320  ℕ₀cn0 12529  ...cfz 13561  ♯chash 14394  Vtxcvtx 29437  iEdgciedg 29438  USGraphcusgr 29593  StarGrcstgr 48852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-xnn0 12603  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13562  df-hash 14395  df-struct 17241  df-slot 17276  df-ndx 17288  df-base 17304  df-edgf 29430  df-vtx 29439  df-iedg 29440  df-usgr 29595  df-stgr 48853

This theorem is used by:  isubgr3stgrlem8  48874  isubgr3stgr  48876