Theorem vdgrfval 26422

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

Assertion

Ref	Expression
vdgrfval	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))

Distinct variable groups:   𝑥,𝑢,𝐴   𝑢,𝐸,𝑥   𝑢,𝑉,𝑥   𝑢,𝑊,𝑥   𝑢,𝑋,𝑥

Proof of Theorem vdgrfval

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

Step	Hyp	Ref	Expression
1		df-vdgr 26421	. . 3 ⊢ VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
2	1	a1i 11	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))))
3		simprl 790	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → 𝑣 = 𝑉)
4		simprr 792	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → 𝑒 = 𝐸)
5	4	dmeqd 5248	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → dom 𝑒 = dom 𝐸)
6		simpl2 1058	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → 𝐸 Fn 𝐴)
7		fndm 5904	. . . . . . . 8 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
8	6, 7	syl 17	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → dom 𝐸 = 𝐴)
9	5, 8	eqtrd 2644	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → dom 𝑒 = 𝐴)
10	4	fveq1d 6105	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑒‘𝑥) = (𝐸‘𝑥))
11	10	eleq2d 2673	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑢 ∈ (𝑒‘𝑥) ↔ 𝑢 ∈ (𝐸‘𝑥)))
12	9, 11	rabeqbidv 3168	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)})
13	12	fveq2d 6107	. . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) = (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}))
14	10	eqeq1d 2612	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑒‘𝑥) = {𝑢} ↔ (𝐸‘𝑥) = {𝑢}))
15	9, 14	rabeqbidv 3168	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})
16	15	fveq2d 6107	. . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}) = (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))
17	13, 16	oveq12d 6567	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))
18	3, 17	mpteq12dv 4663	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))
19		elex 3185	. . 3 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
20	19	3ad2ant1 1075	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝑉 ∈ V)
21		fnex 6386	. . 3 ⊢ ((𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝐸 ∈ V)
22	21	3adant1 1072	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝐸 ∈ V)
23		mptexg 6389	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))) ∈ V)
24	23	3ad2ant1 1075	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))) ∈ V)
25	2, 18, 20, 22, 24	ovmpt2d 6686	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {csn 4125   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   +_𝑒 cxad 11820  #chash 12979   VDeg cvdg 26420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-vdgr 26421

This theorem is referenced by:  vdgrval  26423  vdgrf  26425  vdgrfif  26426