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Theorem vtxdgop 26422
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 4962 . . 3 ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
2 fvex 6239 . . . . . 6 (Vtx‘𝐺) ∈ V
3 fvex 6239 . . . . . 6 (iEdg‘𝐺) ∈ V
42, 3opvtxfvi 25934 . . . . 5 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
54eqcomi 2660 . . . 4 (Vtx‘𝐺) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
62, 3opiedgfvi 25935 . . . . 5 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
76eqcomi 2660 . . . 4 (iEdg‘𝐺) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8 eqid 2651 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
95, 7, 8vtxdgfval 26419 . . 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 6693 . . 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 2651 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
14 eqid 2651 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
1513, 14, 8vtxdgfval 26419 . 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 2696 1 (𝐺𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  {csn 4210  cop 4216  cmpt 4762  dom cdm 5143  cfv 5926  (class class class)co 6690   +𝑒 cxad 11982  #chash 13157  Vtxcvtx 25919  iEdgciedg 25920  VtxDegcvtxdg 26417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211  df-vtx 25921  df-iedg 25922  df-vtxdg 26418
This theorem is referenced by:  finsumvtxdg2size  26502
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