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Theorem vtxdg0e 29507
Description: The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph (see also the induction steps vdegp1ai 29569, vdegp1bi 29570 and vdegp1ci 29571). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
Hypotheses
Ref Expression
vtxdgf.v 𝑉 = (Vtx‘𝐺)
vtxdg0e.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
vtxdg0e ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Proof of Theorem vtxdg0e
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vtxdg0e.i . . . . 5 𝐼 = (iEdg‘𝐺)
21eqeq1i 2740 . . . 4 (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3 dmeq 5917 . . . . . 6 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4 dm0 5934 . . . . . 6 dom ∅ = ∅
53, 4eqtrdi 2791 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6 0fi 9081 . . . . 5 ∅ ∈ Fin
75, 6eqeltrdi 2847 . . . 4 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
82, 7sylbi 217 . . 3 (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9 simpl 482 . . 3 ((𝑈𝑉𝐼 = ∅) → 𝑈𝑉)
10 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
11 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
12 eqid 2735 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1310, 11, 12vtxdgfival 29502 . . 3 ((dom (iEdg‘𝐺) ∈ Fin ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
148, 9, 13syl2an2 686 . 2 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
152, 5sylbi 217 . . . . 5 (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
1615adantl 481 . . . 4 ((𝑈𝑉𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17 rabeq 3448 . . . . . . . 8 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18 rab0 4392 . . . . . . . 8 {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
1917, 18eqtrdi 2791 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
2019fveq2d 6911 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21 hash0 14403 . . . . . 6 (♯‘∅) = 0
2220, 21eqtrdi 2791 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23 rabeq 3448 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
2423fveq2d 6911 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25 rab0 4392 . . . . . . . 8 {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
2625fveq2i 6910 . . . . . . 7 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
2726, 21eqtri 2763 . . . . . 6 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
2824, 27eqtrdi 2791 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
2922, 28oveq12d 7449 . . . 4 (dom (iEdg‘𝐺) = ∅ → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = (0 + 0))
3016, 29syl 17 . . 3 ((𝑈𝑉𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = (0 + 0))
31 00id 11434 . . 3 (0 + 0) = 0
3230, 31eqtrdi 2791 . 2 ((𝑈𝑉𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
3314, 32eqtrd 2775 1 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  c0 4339  {csn 4631  dom cdm 5689  cfv 6563  (class class class)co 7431  Fincfn 8984  0cc0 11153   + caddc 11156  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  VtxDegcvtxdg 29498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-xadd 13153  df-fz 13545  df-hash 14367  df-vtxdg 29499
This theorem is referenced by:  vtxduhgr0e  29511  0edg0rgr  29605  eupth2lemb  30266  konigsberglem1  30281  konigsberglem2  30282  konigsberglem3  30283
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