Theorem vtxdgop 16216

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 15942	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V)
2		iedgex 15943	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘𝐺) ∈ V)
3		opexg 4326	. . . . 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 2231	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
6		eqid 2231	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
7		eqid 2231	. . . . 5 ⊢ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	5, 6, 7	vtxdgfval 16212	. . . 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 15946	. . . . 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 15949	. . . . . . . . 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 4939	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15	13	fveq1d 5650	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
16	15	eleq2d 2301	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
17	14, 16	rabeqbidv 2798	. . . . . 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 5652	. . . . 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 2240	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
20	14, 19	rabeqbidv 2798	. . . . . 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 5652	. . . . 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 6046	. . . 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 4176	. . 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 2264	. 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 6031	. . 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 2231	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
28		eqid 2231	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
29		eqid 2231	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
30	27, 28, 29	vtxdgfval 16212	. 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 2275	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  {crab 2515  Vcvv 2803  {csn 3673  ⟨cop 3676   ↦ cmpt 4155  dom cdm 4731  ‘cfv 5333  (class class class)co 6028   +_𝑒 cxad 10049  ♯chash 11083  Vtxcvtx 15936  iEdgciedg 15937  VtxDegcvtxdg 16210

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-vtxdg 16211

This theorem is referenced by: (None)