Theorem vtxdgval 27510

Description: The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
vtxdgval	⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,𝑈

Allowed substitution hints:   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem vtxdgval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgval.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	1vgrex 27047	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐺 ∈ V)
3		vtxdgval.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
4		vtxdgval.a	. . . . 5 ⊢ 𝐴 = dom 𝐼
5	1, 3, 4	vtxdgfval 27509	. . . 4 ⊢ (𝐺 ∈ V → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
6	2, 5	syl 17	. . 3 ⊢ (𝑈 ∈ 𝑉 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
7	6	fveq1d 6697	. 2 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈))
8		eleq1 2818	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐼‘𝑥) ↔ 𝑈 ∈ (𝐼‘𝑥)))
9	8	rabbidv 3380	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})
10	9	fveq2d 6699	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
11		sneq 4537	. . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈})
12	11	eqeq2d 2747	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐼‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑈}))
13	12	rabbidv 3380	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})
14	13	fveq2d 6699	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))
15	10, 14	oveq12d 7209	. . 3 ⊢ (𝑢 = 𝑈 → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
16		eqid 2736	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))
17		ovex 7224	. . 3 ⊢ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) ∈ V
18	15, 16, 17	fvmpt 6796	. 2 ⊢ (𝑈 ∈ 𝑉 → ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
19	7, 18	eqtrd 2771	1 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  {crab 3055  Vcvv 3398  {csn 4527   ↦ cmpt 5120  dom cdm 5536  ‘cfv 6358  (class class class)co 7191   +_𝑒 cxad 12667  ♯chash 13861  Vtxcvtx 27041  iEdgciedg 27042  VtxDegcvtxdg 27507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-vtxdg 27508

This theorem is referenced by:  vtxdgfival  27511  vtxdun  27523  vtxdlfgrval  27527  vtxd0nedgb  27530  vtxdushgrfvedg  27532  vtxdginducedm1  27585