Theorem stgrusgra 49260

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 6883	. . . . 5 ⊢ ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}⟶1-1onto{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}
2		f1of1 6842	. . . . 5 ⊢ (( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}):{𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}⟶1-1onto{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 6905	. . . . . . . . . . 11 ⊢ (𝑘 = {0, 𝑥} → (♯‘𝑘) = (♯‘{0, 𝑥}))
6		0red 11335	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 ∈ ℝ)
7		elfznn 13711	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
8	7	nngt0d 12411	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...𝑁) → 0 < 𝑥)
9	6, 8	ltned 11470	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑁) → 0 ≠ 𝑥)
10		c0ex 11324	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
11		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12	10, 11	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V ∧ 𝑥 ∈ V)
13		hashprg 14563	. . . . . . . . . . . . . 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 2820	. . . . . . . . . 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 3167	. . . . . . 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 2767	. . . . . . . 8 ⊢ (𝑒 = 𝑘 → (𝑒 = {0, 𝑥} ↔ 𝑘 = {0, 𝑥}))
23	22	rexbidv 3190	. . . . . . 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 6914	. . . . . . 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 3919	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} ⊆ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
29		f1ss 6804	. . . 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 49256	. . . 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 6055	. . . . 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 2814	. . . 4 ⊢ (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
35		stgrvtx 49255	. . . . . 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 3431	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
38	31, 34, 37	f1eq123d 6835	. . 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 6918	. . 3 ⊢ (StarGr‘𝑁) ∈ V
41		eqid 2763	. . . 4 ⊢ (Vtx‘(StarGr‘𝑁)) = (Vtx‘(StarGr‘𝑁))
42		eqid 2763	. . . 4 ⊢ (iEdg‘(StarGr‘𝑁)) = (iEdg‘(StarGr‘𝑁))
43	41, 42	isusgrs 29764	. . 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 2147   ≠ wne 2959  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  {cpr 4590   I cid 5549  dom cdm 5655   ↾ cres 5657  ⟶1-1wf1 6552  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  0cc0 11224  1c1 11225  2c2 12421  ℕ₀cn0 12630  ...cfz 13663  ♯chash 14498  Vtxcvtx 29601  iEdgciedg 29602  USGraphcusgr 29757  StarGrcstgr 49252

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-xnn0 12704  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-hash 14499  df-struct 17351  df-slot 17386  df-ndx 17398  df-base 17414  df-edgf 29594  df-vtx 29603  df-iedg 29604  df-usgr 29759  df-stgr 49253

This theorem is used by:  isubgr3stgrlem8  49274  isubgr3stgr  49276