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Theorem hashnbgravdg 26440
 Description: The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 26439. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
hashnbgravdg ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))

Proof of Theorem hashnbgravdg
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgraf1o 25976 . . . 4 ((𝑉 USGrph 𝐸𝑈𝑉) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑈)–1-1-onto→{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})
2 hasheqf1o 12999 . . . 4 (((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}) ↔ ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑈)–1-1-onto→{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))
31, 2syl5ibr 235 . . 3 (((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
4 edgusgranbfin 25979 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ↔ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin))
5 pm2.24 120 . . . . . 6 ({𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → (¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
64, 5syl6bi 242 . . . . 5 ((𝑉 USGrph 𝐸𝑈𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → (¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))))
76com3l 87 . . . 4 ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → (¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))))
87imp 444 . . 3 (((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
9 pm2.24 120 . . . . . 6 ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → (¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
104, 9syl6bir 243 . . . . 5 ((𝑉 USGrph 𝐸𝑈𝑉) → ({𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → (¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))))
1110com13 86 . . . 4 (¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin → ({𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))))
1211imp 444 . . 3 ((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
13 ovex 6577 . . . . . . 7 (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ V
1413a1i 11 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ V)
15 simpl 472 . . . . . 6 ((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin)
16 hashinf 12984 . . . . . 6 (((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ V ∧ ¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = +∞)
1714, 15, 16syl2anr 494 . . . . 5 (((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑈𝑉)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = +∞)
18 usgrav 25867 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
1918simprd 478 . . . . . . . 8 (𝑉 USGrph 𝐸𝐸 ∈ V)
20 dmexg 6989 . . . . . . . 8 (𝐸 ∈ V → dom 𝐸 ∈ V)
21 rabexg 4739 . . . . . . . 8 (dom 𝐸 ∈ V → {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ V)
2219, 20, 213syl 18 . . . . . . 7 (𝑉 USGrph 𝐸 → {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ V)
2322adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉) → {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ V)
24 simpr 476 . . . . . 6 ((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin)
25 hashinf 12984 . . . . . 6 (({𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ V ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}) = +∞)
2623, 24, 25syl2anr 494 . . . . 5 (((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑈𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}) = +∞)
2717, 26eqtr4d 2647 . . . 4 (((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑈𝑉)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))
2827ex 449 . . 3 ((¬ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ∧ ¬ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin) → ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)})))
293, 8, 12, 284cases 987 . 2 ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))
30 vdusgraval 26434 . 2 ((𝑉 USGrph 𝐸𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))
3129, 30eqtr4d 2647 1 ((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  +∞cpnf 9950  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946   VDeg cvdg 26420 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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421 This theorem is referenced by:  nbhashnn0  26441  usgrauvtxvdbi  26447  cusgraisrusgra  26465  rusgraprop2  26469  frgrancvvdgeq  26570  usgreghash2spotv  26593
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