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Theorem vtxdg0e 27173
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 27235, vdegp1bi 27236 and vdegp1ci 27237). (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 2830 . . . 4 (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3 dmeq 5770 . . . . . 6 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4 dm0 5788 . . . . . 6 dom ∅ = ∅
53, 4syl6eq 2876 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6 0fin 8738 . . . . 5 ∅ ∈ Fin
75, 6syl6eqel 2925 . . . 4 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
82, 7sylbi 218 . . 3 (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9 simpl 483 . . 3 ((𝑈𝑉𝐼 = ∅) → 𝑈𝑉)
10 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
11 eqid 2825 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
12 eqid 2825 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1310, 11, 12vtxdgfival 27168 . . 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 682 . 2 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
152, 5sylbi 218 . . . . 5 (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
1615adantl 482 . . . 4 ((𝑈𝑉𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17 rabeq 3488 . . . . . . . 8 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18 rab0 4340 . . . . . . . 8 {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
1917, 18syl6eq 2876 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
2019fveq2d 6670 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21 hash0 13721 . . . . . 6 (♯‘∅) = 0
2220, 21syl6eq 2876 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23 rabeq 3488 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
2423fveq2d 6670 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25 rab0 4340 . . . . . . . 8 {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
2625fveq2i 6669 . . . . . . 7 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
2726, 21eqtri 2848 . . . . . 6 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
2824, 27syl6eq 2876 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
2922, 28oveq12d 7169 . . . 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 10807 . . 3 (0 + 0) = 0
3230, 31syl6eq 2876 . 2 ((𝑈𝑉𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
3314, 32eqtrd 2860 1 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {crab 3146  c0 4294  {csn 4563  dom cdm 5553  cfv 6351  (class class class)co 7151  Fincfn 8501  0cc0 10529   + caddc 10532  chash 13683  Vtxcvtx 26698  iEdgciedg 26699  VtxDegcvtxdg 27164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-xadd 12501  df-fz 12886  df-hash 13684  df-vtxdg 27165
This theorem is referenced by:  vtxduhgr0e  27177  0edg0rgr  27271  eupth2lemb  27933  konigsberglem1  27948  konigsberglem2  27949  konigsberglem3  27950
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