Theorem vtxdgfval 27737

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 27736	. 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 6769	. . . 4 ⊢ (Vtx‘𝑔) ∈ V
3		fvex 6769	. . . 4 ⊢ (iEdg‘𝑔) ∈ V
4		simpl 482	. . . . 5 ⊢ ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
5		dmeq 5801	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
6		fveq1 6755	. . . . . . . . . 10 ⊢ (𝑒 = (iEdg‘𝑔) → (𝑒‘𝑥) = ((iEdg‘𝑔)‘𝑥))
7	6	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
8	5, 7	rabeqbidv 3410	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
9	8	fveq2d 6760	. . . . . . 7 ⊢ (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
10	6	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝑔) → ((𝑒‘𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
11	5, 10	rabeqbidv 3410	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
12	11	fveq2d 6760	. . . . . . 7 ⊢ (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
13	9, 12	oveq12d 7273	. . . . . 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 481	. . . . 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 5161	. . . 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 3870	. . 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 6756	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18		vtxdgfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
19	17, 18	eqtr4di 2797	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
20		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
21	20	dmeqd 5803	. . . . . . . . 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 5802	. . . . . . . . . 10 ⊢ dom 𝐼 = dom (iEdg‘𝐺)
25	22, 24	eqtri 2766	. . . . . . . . 9 ⊢ 𝐴 = dom (iEdg‘𝐺)
26	21, 25	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
27	20, 23	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
28	27	fveq1d 6758	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼‘𝑥))
29	28	eleq2d 2824	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼‘𝑥)))
30	26, 29	rabeqbidv 3410	. . . . . . 7 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)})
31	30	fveq2d 6760	. . . . . 6 ⊢ (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}))
32	28	eqeq1d 2740	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑢}))
33	26, 32	rabeqbidv 3410	. . . . . . 7 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})
34	33	fveq2d 6760	. . . . . 6 ⊢ (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))
35	31, 34	oveq12d 7273	. . . . 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 5161	. . . 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 481	. . 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 2791	. 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 3440	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
40	18	fvexi 6770	. . 3 ⊢ 𝑉 ∈ V
41		mptexg 7079	. . 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 6865	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  ⦋csb 3828  {csn 4558   ↦ cmpt 5153  dom cdm 5580  ‘cfv 6418  (class class class)co 7255   +_𝑒 cxad 12775  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  VtxDegcvtxdg 27735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-vtxdg 27736

This theorem is referenced by:  vtxdgval  27738  vtxdgop  27740  vtxdgf  27741  vtxdeqd  27747