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

Theorem edgusgrnbfin 40601
 Description: The number of neighbors of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nbusgrf1o.v 𝑉 = (Vtx‘𝐺)
nbusgrf1o.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
edgusgrnbfin ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐺 NeighbVtx 𝑈) ∈ Fin ↔ {𝑒𝐸𝑈𝑒} ∈ Fin))
Distinct variable groups:   𝑒,𝐸   𝑈,𝑒
Allowed substitution hints:   𝐺(𝑒)   𝑉(𝑒)

Proof of Theorem edgusgrnbfin
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 nbusgrf1o.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbusgrf1o.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbusgrf1o 40599 . . 3 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒})
4 f1ofo 6057 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → 𝑓:(𝐺 NeighbVtx 𝑈)–onto→{𝑒𝐸𝑈𝑒})
5 fofi 8135 . . . . . 6 (((𝐺 NeighbVtx 𝑈) ∈ Fin ∧ 𝑓:(𝐺 NeighbVtx 𝑈)–onto→{𝑒𝐸𝑈𝑒}) → {𝑒𝐸𝑈𝑒} ∈ Fin)
65expcom 450 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–onto→{𝑒𝐸𝑈𝑒} → ((𝐺 NeighbVtx 𝑈) ∈ Fin → {𝑒𝐸𝑈𝑒} ∈ Fin))
74, 6syl 17 . . . 4 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → ((𝐺 NeighbVtx 𝑈) ∈ Fin → {𝑒𝐸𝑈𝑒} ∈ Fin))
87exlimiv 1845 . . 3 (∃𝑓 𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → ((𝐺 NeighbVtx 𝑈) ∈ Fin → {𝑒𝐸𝑈𝑒} ∈ Fin))
93, 8syl 17 . 2 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐺 NeighbVtx 𝑈) ∈ Fin → {𝑒𝐸𝑈𝑒} ∈ Fin))
10 f1of1 6049 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → 𝑓:(𝐺 NeighbVtx 𝑈)–1-1→{𝑒𝐸𝑈𝑒})
11 f1fi 8136 . . . . . 6 (({𝑒𝐸𝑈𝑒} ∈ Fin ∧ 𝑓:(𝐺 NeighbVtx 𝑈)–1-1→{𝑒𝐸𝑈𝑒}) → (𝐺 NeighbVtx 𝑈) ∈ Fin)
1211expcom 450 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1→{𝑒𝐸𝑈𝑒} → ({𝑒𝐸𝑈𝑒} ∈ Fin → (𝐺 NeighbVtx 𝑈) ∈ Fin))
1310, 12syl 17 . . . 4 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → ({𝑒𝐸𝑈𝑒} ∈ Fin → (𝐺 NeighbVtx 𝑈) ∈ Fin))
1413exlimiv 1845 . . 3 (∃𝑓 𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → ({𝑒𝐸𝑈𝑒} ∈ Fin → (𝐺 NeighbVtx 𝑈) ∈ Fin))
153, 14syl 17 . 2 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ({𝑒𝐸𝑈𝑒} ∈ Fin → (𝐺 NeighbVtx 𝑈) ∈ Fin))
169, 15impbid 201 1 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐺 NeighbVtx 𝑈) ∈ Fin ↔ {𝑒𝐸𝑈𝑒} ∈ Fin))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   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-fal 1481  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-uspgr 40380  df-usgr 40381  df-nbgr 40554 This theorem is referenced by:  nbusgrfi  40602
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