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Theorem vtxdg0e 28422
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 28484, vdegp1bi 28485 and vdegp1ci 28486). (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 2741 . . . 4 (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3 dmeq 5859 . . . . . 6 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4 dm0 5876 . . . . . 6 dom ∅ = ∅
53, 4eqtrdi 2792 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6 0fin 9115 . . . . 5 ∅ ∈ Fin
75, 6eqeltrdi 2846 . . . 4 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
82, 7sylbi 216 . . 3 (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9 simpl 483 . . 3 ((𝑈𝑉𝐼 = ∅) → 𝑈𝑉)
10 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
11 eqid 2736 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
12 eqid 2736 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1310, 11, 12vtxdgfival 28417 . . 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 684 . 2 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
152, 5sylbi 216 . . . . 5 (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
1615adantl 482 . . . 4 ((𝑈𝑉𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17 rabeq 3421 . . . . . . . 8 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18 rab0 4342 . . . . . . . 8 {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
1917, 18eqtrdi 2792 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
2019fveq2d 6846 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21 hash0 14267 . . . . . 6 (♯‘∅) = 0
2220, 21eqtrdi 2792 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23 rabeq 3421 . . . . . . 7 (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
2423fveq2d 6846 . . . . . 6 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25 rab0 4342 . . . . . . . 8 {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
2625fveq2i 6845 . . . . . . 7 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
2726, 21eqtri 2764 . . . . . 6 (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
2824, 27eqtrdi 2792 . . . . 5 (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
2922, 28oveq12d 7375 . . . 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 11330 . . 3 (0 + 0) = 0
3230, 31eqtrdi 2792 . 2 ((𝑈𝑉𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
3314, 32eqtrd 2776 1 ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3407  c0 4282  {csn 4586  dom cdm 5633  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051   + caddc 11054  chash 14230  Vtxcvtx 27947  iEdgciedg 27948  VtxDegcvtxdg 28413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-xadd 13034  df-fz 13425  df-hash 14231  df-vtxdg 28414
This theorem is referenced by:  vtxduhgr0e  28426  0edg0rgr  28520  eupth2lemb  29181  konigsberglem1  29196  konigsberglem2  29197  konigsberglem3  29198
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