Theorem vtxdgval 28714

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 28251	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐺 ∈ V)
3		vtxdgval.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
4		vtxdgval.a	. . . . 5 ⊢ 𝐴 = dom 𝐼
5	1, 3, 4	vtxdgfval 28713	. . . 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 6890	. 2 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈))
8		eleq1 2821	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐼‘𝑥) ↔ 𝑈 ∈ (𝐼‘𝑥)))
9	8	rabbidv 3440	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})
10	9	fveq2d 6892	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
11		sneq 4637	. . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈})
12	11	eqeq2d 2743	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐼‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑈}))
13	12	rabbidv 3440	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})
14	13	fveq2d 6892	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))
15	10, 14	oveq12d 7423	. . 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 2732	. . 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 7438	. . 3 ⊢ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) ∈ V
18	15, 16, 17	fvmpt 6995	. 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 2772	1 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  {csn 4627   ↦ cmpt 5230  dom cdm 5675  ‘cfv 6540  (class class class)co 7405   +_𝑒 cxad 13086  ♯chash 14286  Vtxcvtx 28245  iEdgciedg 28246  VtxDegcvtxdg 28711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-vtxdg 28712

This theorem is referenced by:  vtxdgfival  28715  vtxdun  28727  vtxdlfgrval  28731  vtxd0nedgb  28734  vtxdushgrfvedg  28736  vtxdginducedm1  28789