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Theorem vtxdgop 27270
 Description: The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
Assertion
Ref Expression
vtxdgop (𝐺𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Proof of Theorem vtxdgop
Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5322 . . 3 ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
2 fvex 6659 . . . . . 6 (Vtx‘𝐺) ∈ V
3 fvex 6659 . . . . . 6 (iEdg‘𝐺) ∈ V
42, 3opvtxfvi 26812 . . . . 5 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
54eqcomi 2807 . . . 4 (Vtx‘𝐺) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
62, 3opiedgfvi 26813 . . . . 5 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
76eqcomi 2807 . . . 4 (iEdg‘𝐺) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8 eqid 2798 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
95, 7, 8vtxdgfval 27267 . . 3 (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V → (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
101, 9mp1i 13 . 2 (𝐺𝑊 → (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
11 df-ov 7139 . . 3 ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
1211a1i 11 . 2 (𝐺𝑊 → ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩))
13 eqid 2798 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
14 eqid 2798 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
1513, 14, 8vtxdgfval 27267 . 2 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
1610, 12, 153eqtr4rd 2844 1 (𝐺𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  {csn 4525  ⟨cop 4531   ↦ cmpt 5111  dom cdm 5520  ‘cfv 6325  (class class class)co 7136   +𝑒 cxad 12496  ♯chash 13689  Vtxcvtx 26799  iEdgciedg 26800  VtxDegcvtxdg 27265 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-1st 7674  df-2nd 7675  df-vtx 26801  df-iedg 26802  df-vtxdg 27266 This theorem is referenced by:  finsumvtxdg2size  27350
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