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Theorem vtxdg0e 26273
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 26335, vdegp1bi 26336 and vdegp1ci 26337). (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 2626 . . . 4 (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3 dmeq 5289 . . . . . 6 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4 dm0 5304 . . . . . 6 dom ∅ = ∅
53, 4syl6eq 2671 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6 0fin 8140 . . . . 5 ∅ ∈ Fin
75, 6syl6eqel 2706 . . . 4 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
82, 7sylbi 207 . . 3 (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9 simpl 473 . . 3 ((𝑈𝑉𝐼 = ∅) → 𝑈𝑉)
10 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
11 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
12 eqid 2621 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1310, 11, 12vtxdgfival 26269 . . 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 874 . 2 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
152, 5sylbi 207 . . . . 5 (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
1615adantl 482 . . . 4 ((𝑈𝑉𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17 rabeq 3182 . . . . . . . 8 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18 rab0 3934 . . . . . . . 8 {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
1917, 18syl6eq 2671 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
2019fveq2d 6157 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘∅))
21 hash0 13106 . . . . . 6 (#‘∅) = 0
2220, 21syl6eq 2671 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23 rabeq 3182 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
2423fveq2d 6157 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25 rab0 3934 . . . . . . . 8 {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
2625fveq2i 6156 . . . . . . 7 (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘∅)
2726, 21eqtri 2643 . . . . . 6 (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
2824, 27syl6eq 2671 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
2922, 28oveq12d 6628 . . . 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 10163 . . 3 (0 + 0) = 0
3230, 31syl6eq 2671 . 2 ((𝑈𝑉𝐼 = ∅) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
3314, 32eqtrd 2655 1 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  c0 3896  {csn 4153  dom cdm 5079  cfv 5852  (class class class)co 6610  Fincfn 7907  0cc0 9888   + caddc 9891  #chash 13065  Vtxcvtx 25791  iEdgciedg 25792  VtxDegcvtxdg 26265
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-xadd 11899  df-fz 12277  df-hash 13066  df-vtxdg 26266
This theorem is referenced by:  vtxduhgr0e  26277  0edg0rgr  26355  eupth2lemb  26980  konigsberglem1  26997  konigsberglem2  26998  konigsberglem3  26999
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