Theorem vtxdgval 29234

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 28770	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐺 ∈ V)
3		vtxdgval.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
4		vtxdgval.a	. . . . 5 ⊢ 𝐴 = dom 𝐼
5	1, 3, 4	vtxdgfval 29233	. . . 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 6887	. 2 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈))
8		eleq1 2815	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐼‘𝑥) ↔ 𝑈 ∈ (𝐼‘𝑥)))
9	8	rabbidv 3434	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})
10	9	fveq2d 6889	. . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
11		sneq 4633	. . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈})
12	11	eqeq2d 2737	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐼‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑈}))
13	12	rabbidv 3434	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})
14	13	fveq2d 6889	. . . 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 2726	. . 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 6992	. 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 2766	1 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468  {csn 4623   ↦ cmpt 5224  dom cdm 5669  ‘cfv 6537  (class class class)co 7405   +_𝑒 cxad 13096  ♯chash 14295  Vtxcvtx 28764  iEdgciedg 28765  VtxDegcvtxdg 29231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-vtxdg 29232

This theorem is referenced by:  vtxdgfival  29235  vtxdun  29247  vtxdlfgrval  29251  vtxd0nedgb  29254  vtxdushgrfvedg  29256  vtxdginducedm1  29309