Theorem vtxdgop 29557

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 5403	. . 3 ⊢ ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
2		fvex 6840	. . . . . 6 ⊢ (Vtx‘𝐺) ∈ V
3		fvex 6840	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
4	2, 3	opvtxfvi 29096	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
5	4	eqcomi 2748	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
6	2, 3	opiedgfvi 29097	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
7	6	eqcomi 2748	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8		eqid 2739	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
9	5, 7, 8	vtxdgfval 29554	. . 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 13	. 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 7359	. . 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 2739	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14		eqid 2739	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15	13, 14, 8	vtxdgfval 29554	. 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 2785	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {crab 3391  Vcvv 3431  {csn 4555  ⟨cop 4561   ↦ cmpt 5153  dom cdm 5618  ‘cfv 6485  (class class class)co 7356   +_𝑒 cxad 13052  ♯chash 14283  Vtxcvtx 29083  iEdgciedg 29084  VtxDegcvtxdg 29552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-1st 7931  df-2nd 7932  df-vtx 29085  df-iedg 29086  df-vtxdg 29553

This theorem is referenced by:  finsumvtxdg2size  29637