Theorem vtxdgop 16098

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		vtxex 15859	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V)
2		iedgex 15860	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘𝐺) ∈ V)
3		opexg 4318	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (𝐺 ∈ 𝑊 → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
5		eqid 2229	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
6		eqid 2229	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
7		eqid 2229	. . . . 5 ⊢ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	5, 6, 7	vtxdgfval 16094	. . . 4 ⊢ (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V → (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (𝑢 ∈ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}}))))
9	4, 8	syl 14	. . 3 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (𝑢 ∈ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}}))))
10		opvtxfv 15863	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
11	1, 2, 10	syl2anc 411	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
12		opiedgfv 15866	. . . . . . . . 9 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	1, 2, 12	syl2anc 411	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
14	13	dmeqd 4931	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15	13	fveq1d 5637	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
16	15	eleq2d 2299	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
17	14, 16	rabeqbidv 2795	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → {𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
18	17	fveq2d 5639	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
19	15	eqeq1d 2238	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
20	14, 19	rabeqbidv 2795	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → {𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
21	20	fveq2d 5639	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
22	18, 21	oveq12d 6031	. . . 4 ⊢ (𝐺 ∈ 𝑊 → ((♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))
23	11, 22	mpteq12dv 4169	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ 𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
24	9, 23	eqtrd 2262	. 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}}))))
25		df-ov 6016	. . 3 ⊢ ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
26	25	a1i 9	. 2 ⊢ (𝐺 ∈ 𝑊 → ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩))
27		eqid 2229	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
28		eqid 2229	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
29		eqid 2229	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
30	27, 28, 29	vtxdgfval 16094	. 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}}))))
31	24, 26, 30	3eqtr4rd 2273	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  {crab 2512  Vcvv 2800  {csn 3667  ⟨cop 3670   ↦ cmpt 4148  dom cdm 4723  ‘cfv 5324  (class class class)co 6013   +_𝑒 cxad 9995  ♯chash 11027  Vtxcvtx 15853  iEdgciedg 15854  VtxDegcvtxdg 16092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8342  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-dec 9602  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-vtxdg 16093

This theorem is referenced by: (None)