Theorem vtxdgop 16051

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 15834	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V)
2		iedgex 15835	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘𝐺) ∈ V)
3		opexg 4314	. . . . 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 16047	. . . 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 15838	. . . . 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 15841	. . . . . . . . 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 4925	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15	13	fveq1d 5631	. . . . . . . 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 2794	. . . . . 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 5633	. . . . 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 2794	. . . . . 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 5633	. . . . 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 6025	. . . 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 4166	. . 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 6010	. . 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 16047	. 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 2799  {csn 3666  ⟨cop 3669   ↦ cmpt 4145  dom cdm 4719  ‘cfv 5318  (class class class)co 6007   +_𝑒 cxad 9978  ♯chash 11009  Vtxcvtx 15828  iEdgciedg 15829  VtxDegcvtxdg 16045

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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-vtxdg 16046

This theorem is referenced by: (None)