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Theorem edgusgrnbfin 26156
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 26154 . . 3 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒})
4 f1ofo 6103 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → 𝑓:(𝐺 NeighbVtx 𝑈)–onto→{𝑒𝐸𝑈𝑒})
5 fofi 8197 . . . . . 6 (((𝐺 NeighbVtx 𝑈) ∈ Fin ∧ 𝑓:(𝐺 NeighbVtx 𝑈)–onto→{𝑒𝐸𝑈𝑒}) → {𝑒𝐸𝑈𝑒} ∈ Fin)
65expcom 451 . . . . 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 1860 . . 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 6095 . . . . 5 (𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒} → 𝑓:(𝐺 NeighbVtx 𝑈)–1-1→{𝑒𝐸𝑈𝑒})
11 f1fi 8198 . . . . . 6 (({𝑒𝐸𝑈𝑒} ∈ Fin ∧ 𝑓:(𝐺 NeighbVtx 𝑈)–1-1→{𝑒𝐸𝑈𝑒}) → (𝐺 NeighbVtx 𝑈) ∈ Fin)
1211expcom 451 . . . . 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 1860 . . 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 202 1 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐺 NeighbVtx 𝑈) ∈ Fin ↔ {𝑒𝐸𝑈𝑒} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  {crab 2916  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Fincfn 7900  Vtxcvtx 25769  Edgcedg 25834   USGraph cusgr 25932   NeighbVtx cnbgr 26105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055  df-edg 25835  df-upgr 25868  df-umgr 25869  df-uspgr 25933  df-usgr 25934  df-nbgr 26109
This theorem is referenced by:  nbusgrfi  26157
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