Theorem vtxdgop 16447

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 16173	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V)
2		iedgex 16174	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘𝐺) ∈ V)
3		opexg 4363	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
4	1, 2, 3	syl2anc 415	. . . 4 ⊢ (𝐺 ∈ 𝑊 → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
5		eqid 2238	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
6		eqid 2238	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
7		eqid 2238	. . . . 5 ⊢ dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	5, 6, 7	vtxdgfval 16443	. . . 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 16177	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
11	1, 2, 10	syl2anc 415	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
12		opiedgfv 16180	. . . . . . . . 9 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	1, 2, 12	syl2anc 415	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
14	13	dmeqd 4978	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15	13	fveq1d 5692	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
16	15	eleq2d 2308	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
17	14, 16	rabeqbidv 2816	. . . . . 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 5694	. . . . 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 2247	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
20	14, 19	rabeqbidv 2816	. . . . . 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 5694	. . . . 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 6093	. . . 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 4208	. . 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 2271	. 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 6078	. . 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 2238	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
28		eqid 2238	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
29		eqid 2238	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
30	27, 28, 29	vtxdgfval 16443	. 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 2282	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821  {csn 3705  ⟨cop 3708   ↦ cmpt 4187  dom cdm 4769  ‘cfv 5372  (class class class)co 6075   +_𝑒 cxad 10151  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  VtxDegcvtxdg 16441

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-vtxdg 16442

This theorem is referenced by: (None)