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

Theorem nbumgrvtx 40568
 Description: The set of neighbors of a vertex in a multigraph. (Contributed by AV, 27-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
nbgrel.v 𝑉 = (Vtx‘𝐺)
nbgrel.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbumgrvtx ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
Distinct variable groups:   𝑛,𝐺   𝑛,𝑁   𝑛,𝑉   𝑛,𝐸

Proof of Theorem nbumgrvtx
Dummy variables 𝑒 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgrel.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbgrel.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 40560 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒})
43adantl 481 . 2 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒})
5 eldifi 3694 . . . . . . . . . 10 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥𝑉)
65adantl 481 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥𝑉)
76adantr 480 . . . . . . . 8 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑒𝐸 ∧ {𝑁, 𝑥} ⊆ 𝑒)) → 𝑥𝑉)
8 umgrupgr 25769 . . . . . . . . . . . . 13 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )
98ad4antr 764 . . . . . . . . . . . 12 (((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑥} ⊆ 𝑒) → 𝐺 ∈ UPGraph )
10 simpr 476 . . . . . . . . . . . . 13 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒𝐸)
1110adantr 480 . . . . . . . . . . . 12 (((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑥} ⊆ 𝑒) → 𝑒𝐸)
12 simpr 476 . . . . . . . . . . . 12 (((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑥} ⊆ 𝑒) → {𝑁, 𝑥} ⊆ 𝑒)
13 simpr 476 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑁𝑉)
1413adantr 480 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑉)
15 vex 3176 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
1615a1i 11 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ∈ V)
17 eldifsn 4260 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑥𝑉𝑥𝑁))
18 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑥𝑁) → 𝑥𝑁)
1918necomd 2837 . . . . . . . . . . . . . . . . 17 ((𝑥𝑉𝑥𝑁) → 𝑁𝑥)
2017, 19sylbi 206 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝑥)
2120adantl 481 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑥)
2214, 16, 213jca 1235 . . . . . . . . . . . . . 14 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑁𝑉𝑥 ∈ V ∧ 𝑁𝑥))
2322adantr 480 . . . . . . . . . . . . 13 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑁𝑉𝑥 ∈ V ∧ 𝑁𝑥))
2423adantr 480 . . . . . . . . . . . 12 (((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑥} ⊆ 𝑒) → (𝑁𝑉𝑥 ∈ V ∧ 𝑁𝑥))
251, 2upgredgpr 25815 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑒𝐸 ∧ {𝑁, 𝑥} ⊆ 𝑒) ∧ (𝑁𝑉𝑥 ∈ V ∧ 𝑁𝑥)) → {𝑁, 𝑥} = 𝑒)
269, 11, 12, 24, 25syl31anc 1321 . . . . . . . . . . 11 (((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑥} ⊆ 𝑒) → {𝑁, 𝑥} = 𝑒)
2726ex 449 . . . . . . . . . 10 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑥} ⊆ 𝑒 → {𝑁, 𝑥} = 𝑒))
28 eleq1 2676 . . . . . . . . . . 11 ({𝑁, 𝑥} = 𝑒 → ({𝑁, 𝑥} ∈ 𝐸𝑒𝐸))
2928biimprd 237 . . . . . . . . . 10 ({𝑁, 𝑥} = 𝑒 → (𝑒𝐸 → {𝑁, 𝑥} ∈ 𝐸))
3027, 10, 29syl6ci 69 . . . . . . . . 9 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑥} ⊆ 𝑒 → {𝑁, 𝑥} ∈ 𝐸))
3130impr 647 . . . . . . . 8 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑒𝐸 ∧ {𝑁, 𝑥} ⊆ 𝑒)) → {𝑁, 𝑥} ∈ 𝐸)
327, 31jca 553 . . . . . . 7 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑒𝐸 ∧ {𝑁, 𝑥} ⊆ 𝑒)) → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸))
3332rexlimdvaa 3014 . . . . . 6 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒 → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)))
3433expimpd 627 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝑉 ∖ {𝑁}) ∧ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒) → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)))
35 simprl 790 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → 𝑥𝑉)
362umgredgne 25816 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ {𝑁, 𝑥} ∈ 𝐸) → 𝑁𝑥)
3736ad2ant2rl 781 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → 𝑁𝑥)
3837necomd 2837 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → 𝑥𝑁)
3935, 38, 17sylanbrc 695 . . . . . . 7 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → 𝑥 ∈ (𝑉 ∖ {𝑁}))
40 simpr 476 . . . . . . . . 9 ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸) → {𝑁, 𝑥} ∈ 𝐸)
4140adantl 481 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → {𝑁, 𝑥} ∈ 𝐸)
42 sseq2 3590 . . . . . . . . 9 (𝑒 = {𝑁, 𝑥} → ({𝑁, 𝑥} ⊆ 𝑒 ↔ {𝑁, 𝑥} ⊆ {𝑁, 𝑥}))
4342adantl 481 . . . . . . . 8 ((((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) ∧ 𝑒 = {𝑁, 𝑥}) → ({𝑁, 𝑥} ⊆ 𝑒 ↔ {𝑁, 𝑥} ⊆ {𝑁, 𝑥}))
44 ssid 3587 . . . . . . . . 9 {𝑁, 𝑥} ⊆ {𝑁, 𝑥}
4544a1i 11 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → {𝑁, 𝑥} ⊆ {𝑁, 𝑥})
4641, 43, 45rspcedvd 3289 . . . . . . 7 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒)
4739, 46jca 553 . . . . . 6 (((𝐺 ∈ UMGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)) → (𝑥 ∈ (𝑉 ∖ {𝑁}) ∧ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒))
4847ex 449 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸) → (𝑥 ∈ (𝑉 ∖ {𝑁}) ∧ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒)))
4934, 48impbid 201 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝑉 ∖ {𝑁}) ∧ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒) ↔ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸)))
50 preq2 4213 . . . . . . 7 (𝑣 = 𝑥 → {𝑁, 𝑣} = {𝑁, 𝑥})
5150sseq1d 3595 . . . . . 6 (𝑣 = 𝑥 → ({𝑁, 𝑣} ⊆ 𝑒 ↔ {𝑁, 𝑥} ⊆ 𝑒))
5251rexbidv 3034 . . . . 5 (𝑣 = 𝑥 → (∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒))
5352elrab 3331 . . . 4 (𝑥 ∈ {𝑣 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒} ↔ (𝑥 ∈ (𝑉 ∖ {𝑁}) ∧ ∃𝑒𝐸 {𝑁, 𝑥} ⊆ 𝑒))
54 preq2 4213 . . . . . 6 (𝑛 = 𝑥 → {𝑁, 𝑛} = {𝑁, 𝑥})
5554eleq1d 2672 . . . . 5 (𝑛 = 𝑥 → ({𝑁, 𝑛} ∈ 𝐸 ↔ {𝑁, 𝑥} ∈ 𝐸))
5655elrab 3331 . . . 4 (𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ↔ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ 𝐸))
5749, 53, 563bitr4g 302 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑣 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒} ↔ 𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}))
5857eqrdv 2608 . 2 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → {𝑣 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑣} ⊆ 𝑒} = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
594, 58eqtrd 2644 1 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   UPGraph cupgr 25747   UMGraph cumgr 25748  Edgcedga 25792   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-nbgr 40554 This theorem is referenced by:  nbumgr  40569  nbusgrvtx  40570  umgr2v2enb1  40742
 Copyright terms: Public domain W3C validator