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Theorem vtxdg0e 40689
 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 (see also vdegp1ai-av 40752, vdegp1ai-av 40752 and vdegp1ai-av 40752). (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 2615 . . . 4 (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3 dmeq 5246 . . . . . 6 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4 dm0 5260 . . . . . 6 dom ∅ = ∅
53, 4syl6eq 2660 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6 0fin 8073 . . . . 5 ∅ ∈ Fin
75, 6syl6eqel 2696 . . . 4 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
82, 7sylbi 206 . . 3 (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9 simpl 472 . . 3 ((𝑈𝑉𝐼 = ∅) → 𝑈𝑉)
10 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
11 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
12 eqid 2610 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1310, 11, 12vtxdgfival 40685 . . 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 871 . 2 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
152, 5sylbi 206 . . . . 5 (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
1615adantl 481 . . . 4 ((𝑈𝑉𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17 rabeq 3166 . . . . . . . 8 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18 rab0 3909 . . . . . . . 8 {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
1917, 18syl6eq 2660 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
2019fveq2d 6107 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘∅))
21 hash0 13019 . . . . . 6 (#‘∅) = 0
2220, 21syl6eq 2660 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23 rabeq 3166 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
2423fveq2d 6107 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25 rab0 3909 . . . . . . . 8 {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
2625fveq2i 6106 . . . . . . 7 (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘∅)
2726, 21eqtri 2632 . . . . . 6 (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
2824, 27syl6eq 2660 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
2922, 28oveq12d 6567 . . . 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 10090 . . 3 (0 + 0) = 0
3230, 31syl6eq 2660 . 2 ((𝑈𝑉𝐼 = ∅) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
3314, 32eqtrd 2644 1 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ∅c0 3874  {csn 4125  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815   + caddc 9818  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  VtxDegcvtxdg 40681 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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-n0 11170  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-vtxdg 40682 This theorem is referenced by:  vtxduhgr0e  40693  0edg0rgr  40772  eupth2lemb  41405  konigsberglem1  41422  konigsberglem2  41423  konigsberglem3  41424
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