Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stgrusgra Structured version   Visualization version   GIF version

Theorem stgrusgra 47861
Description: The star graph SN 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.
StepHypRef 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}})
31, 2mp1i 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 ∈ ℕ0𝑘 ∈ 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 11261 . . . . . . . . . . . . . 14 (𝑥 ∈ (1...𝑁) → 0 ∈ ℝ)
7 elfznn 13589 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
87nngt0d 12312 . . . . . . . . . . . . . 14 (𝑥 ∈ (1...𝑁) → 0 < 𝑥)
96, 8ltned 11394 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑁) → 0 ≠ 𝑥)
10 c0ex 11252 . . . . . . . . . . . . . . 15 0 ∈ V
11 vex 3481 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1210, 11pm3.2i 470 . . . . . . . . . . . . . 14 (0 ∈ V ∧ 𝑥 ∈ V)
13 hashprg 14430 . . . . . . . . . . . . . 14 ((0 ∈ V ∧ 𝑥 ∈ V) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
1412, 13mp1i 13 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑁) → (0 ≠ 𝑥 ↔ (♯‘{0, 𝑥}) = 2))
159, 14mpbid 232 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑁) → (♯‘{0, 𝑥}) = 2)
1615adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (♯‘{0, 𝑥}) = 2)
175, 16sylan9eqr 2796 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (♯‘𝑘) = 2)
184, 17jca 511 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) ∧ 𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2))
1918ex 412 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ 𝒫 (0...𝑁)) ∧ 𝑥 ∈ (1...𝑁)) → (𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
2019rexlimdva 3152 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ 𝒫 (0...𝑁)) → (∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥} → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
2120expimpd 453 . . . . . 6 (𝑁 ∈ ℕ0 → ((𝑘 ∈ 𝒫 (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}) → (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2)))
22 eqeq1 2738 . . . . . . . 8 (𝑒 = 𝑘 → (𝑒 = {0, 𝑥} ↔ 𝑘 = {0, 𝑥}))
2322rexbidv 3176 . . . . . . 7 (𝑒 = 𝑘 → (∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑘 = {0, 𝑥}))
2423elrab 3694 . . . . . 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))
2625elrab 3694 . . . . . 6 (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2} ↔ (𝑘 ∈ 𝒫 (0...𝑁) ∧ (♯‘𝑘) = 2))
2721, 24, 263imtr4g 296 . . . . 5 (𝑁 ∈ ℕ0 → (𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}} → 𝑘 ∈ {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2}))
2827ssrdv 4000 . . . 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})
303, 28, 29syl2anc 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 47857 . . . 4 (𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
3231dmeqd 5918 . . . . 5 (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
33 dmresi 6071 . . . . 5 dom ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}
3432, 33eqtrdi 2790 . . . 4 (𝑁 ∈ ℕ0 → dom (iEdg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
35 stgrvtx 47856 . . . . . 6 (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
3635pweqd 4621 . . . . 5 (𝑁 ∈ ℕ0 → 𝒫 (Vtx‘(StarGr‘𝑁)) = 𝒫 (0...𝑁))
3736rabeqdv 3448 . . . 4 (𝑁 ∈ ℕ0 → {𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ (♯‘𝑒) = 2})
3831, 34, 37f1eq123d 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}))
3930, 38mpbird 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 2734 . . . 4 (Vtx‘(StarGr‘𝑁)) = (Vtx‘(StarGr‘𝑁))
42 eqid 2734 . . . 4 (iEdg‘(StarGr‘𝑁)) = (iEdg‘(StarGr‘𝑁))
4341, 42isusgrs 29187 . . 3 ((StarGr‘𝑁) ∈ V → ((StarGr‘𝑁) ∈ USGraph ↔ (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2}))
4440, 43mp1i 13 . 2 (𝑁 ∈ ℕ0 → ((StarGr‘𝑁) ∈ USGraph ↔ (iEdg‘(StarGr‘𝑁)):dom (iEdg‘(StarGr‘𝑁))–1-1→{𝑒 ∈ 𝒫 (Vtx‘(StarGr‘𝑁)) ∣ (♯‘𝑒) = 2}))
4539, 44mpbird 257 1 (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  𝒫 cpw 4604  {cpr 4632   I cid 5581  dom cdm 5688  cres 5690  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153  2c2 12318  0cn0 12523  ...cfz 13543  chash 14365  Vtxcvtx 29027  iEdgciedg 29028  USGraphcusgr 29180  StarGrcstgr 47853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-hash 14366  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-edgf 29018  df-vtx 29029  df-iedg 29030  df-usgr 29182  df-stgr 47854
This theorem is referenced by:  isubgr3stgrlem8  47875  isubgr3stgr  47877
  Copyright terms: Public domain W3C validator