Theorem vdgrval 26423

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
vdgrval	⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))

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

Proof of Theorem vdgrval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdgrfval 26422	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))
2	1	fveq1d 6105	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → ((𝑉 VDeg 𝐸)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))‘𝑈))
3		eleq1 2676	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐸‘𝑥) ↔ 𝑈 ∈ (𝐸‘𝑥)))
4	3	rabbidv 3164	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)})
5	4	fveq2d 6107	. . . 4 ⊢ (𝑢 = 𝑈 → (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) = (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}))
6		sneq 4135	. . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈})
7	6	eqeq2d 2620	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐸‘𝑥) = {𝑢} ↔ (𝐸‘𝑥) = {𝑈}))
8	7	rabbidv 3164	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})
9	8	fveq2d 6107	. . . 4 ⊢ (𝑢 = 𝑈 → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) = (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))
10	5, 9	oveq12d 6567	. . 3 ⊢ (𝑢 = 𝑈 → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
11		eqid 2610	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))
12		ovex 6577	. . 3 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) ∈ V
13	10, 11, 12	fvmpt 6191	. 2 ⊢ (𝑈 ∈ 𝑉 → ((𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
14	2, 13	sylan9eq 2664	1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{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  {csn 4125   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   +_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:  vdgrfival  26424  vdgr0  26427  vdgrun  26428  vdgr1d  26430  vdgr1b  26431  vdgr1a  26433  vdusgraval  26434