Theorem vtxdgop 29878

Description: The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)

Assertion

Ref	Expression
vtxdgop	⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Proof of Theorem vtxdgop

Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5447	. . 3 ⊢ ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
2		fvex 6898	. . . . . 6 ⊢ (Vtx‘𝐺) ∈ V
3		fvex 6898	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
4	2, 3	opvtxfvi 29414	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
5	4	eqcomi 2774	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
6	2, 3	opiedgfvi 29415	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
7	6	eqcomi 2774	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8		eqid 2765	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
9	5, 7, 8	vtxdgfval 29875	. . 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}}))))
10	1, 9	mp1i 14	. 2 ⊢ (𝐺 ∈ 𝑊 → (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 7422	. . 3 ⊢ ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑊 → ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩))
13		eqid 2765	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14		eqid 2765	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15	13, 14, 8	vtxdgfval 29875	. 2 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
16	10, 12, 15	3eqtr4rd 2811	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457  {csn 4591  ⟨cop 4597   ↦ cmpt 5194  dom cdm 5663  ‘cfv 6540  (class class class)co 7419   +_𝑒 cxad 13151  ♯chash 14384  Vtxcvtx 29401  iEdgciedg 29402  VtxDegcvtxdg 29873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-1st 7992  df-2nd 7993  df-vtx 29403  df-iedg 29404  df-vtxdg 29874

This theorem is used by:  finsumvtxdg2size  29958