Theorem vtxdgfval 26942

Description: The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)

Hypotheses

Ref	Expression
vtxdgfval.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdgfval.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxdgfval.a	⊢ 𝐴 = dom 𝐼

Assertion

Ref	Expression
vtxdgfval	⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))

Distinct variable groups:   𝑥,𝑢   𝑥,𝐴   𝑢,𝐺,𝑥   𝑢,𝑉

Allowed substitution hints:   𝐴(𝑢)   𝐼(𝑥,𝑢)   𝑉(𝑥)   𝑊(𝑥,𝑢)

Proof of Theorem vtxdgfval

Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vtxdg 26941	. 2 ⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
2		fvex 6506	. . . 4 ⊢ (Vtx‘𝑔) ∈ V
3		fvex 6506	. . . 4 ⊢ (iEdg‘𝑔) ∈ V
4		simpl 475	. . . . 5 ⊢ ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
5		dmeq 5615	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
6		fveq1 6492	. . . . . . . . . 10 ⊢ (𝑒 = (iEdg‘𝑔) → (𝑒‘𝑥) = ((iEdg‘𝑔)‘𝑥))
7	6	eleq2d 2845	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
8	5, 7	rabeqbidv 3402	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
9	8	fveq2d 6497	. . . . . . 7 ⊢ (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
10	6	eqeq1d 2774	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → ((𝑒‘𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
11	5, 10	rabeqbidv 3402	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
12	11	fveq2d 6497	. . . . . . 7 ⊢ (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
13	9, 12	oveq12d 6988	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝑔) → ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
14	13	adantl 474	. . . . 5 ⊢ ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
15	4, 14	mpteq12dv 5006	. . . 4 ⊢ ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → (𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))))
16	2, 3, 15	csbie2 3814	. . 3 ⊢ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
17		fveq2 6493	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18		vtxdgfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
19	17, 18	syl6eqr 2826	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
20		fveq2 6493	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
21	20	dmeqd 5617	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom (iEdg‘𝐺))
22		vtxdgfval.a	. . . . . . . . . 10 ⊢ 𝐴 = dom 𝐼
23		vtxdgfval.i	. . . . . . . . . . 11 ⊢ 𝐼 = (iEdg‘𝐺)
24	23	dmeqi 5616	. . . . . . . . . 10 ⊢ dom 𝐼 = dom (iEdg‘𝐺)
25	22, 24	eqtri 2796	. . . . . . . . 9 ⊢ 𝐴 = dom (iEdg‘𝐺)
26	21, 25	syl6eqr 2826	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
27	20, 23	syl6eqr 2826	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
28	27	fveq1d 6495	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼‘𝑥))
29	28	eleq2d 2845	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼‘𝑥)))
30	26, 29	rabeqbidv 3402	. . . . . . 7 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)})
31	30	fveq2d 6497	. . . . . 6 ⊢ (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}))
32	28	eqeq1d 2774	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑢}))
33	26, 32	rabeqbidv 3402	. . . . . . 7 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})
34	33	fveq2d 6497	. . . . . 6 ⊢ (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))
35	31, 34	oveq12d 6988	. . . . 5 ⊢ (𝑔 = 𝐺 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))
36	19, 35	mpteq12dv 5006	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
37	36	adantl 474	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑔 = 𝐺) → (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
38	16, 37	syl5eq 2820	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑔 = 𝐺) → ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
39		elex 3427	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
40	18	fvexi 6507	. . 3 ⊢ 𝑉 ∈ V
41		mptexg 6804	. . 3 ⊢ (𝑉 ∈ V → (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))) ∈ V)
42	40, 41	mp1i 13	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))) ∈ V)
43	1, 38, 39, 42	fvmptd2 6596	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048  {crab 3086  Vcvv 3409  ⦋csb 3782  {csn 4435   ↦ cmpt 5002  dom cdm 5400  ‘cfv 6182  (class class class)co 6970   +_𝑒 cxad 12315  ♯chash 13498  Vtxcvtx 26474  iEdgciedg 26475  VtxDegcvtxdg 26940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-vtxdg 26941

This theorem is referenced by:  vtxdgval  26943  vtxdgop  26945  vtxdgf  26946  vtxdeqd  26952