Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vtxdgval Structured version   Visualization version   GIF version

Theorem vtxdgval 40684
 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 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {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.
StepHypRef Expression
1 vtxdgval.v . . . . 5 𝑉 = (Vtx‘𝐺)
211vgrex 25679 . . . 4 (𝑈𝑉𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdg‘𝐺)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 40683 . . . 4 (𝐺 ∈ V → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
62, 5syl 17 . . 3 (𝑈𝑉 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
76fveq1d 6105 . 2 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈))
8 eleq1 2676 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∈ (𝐼𝑥) ↔ 𝑈 ∈ (𝐼𝑥)))
98rabbidv 3164 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴𝑢 ∈ (𝐼𝑥)} = {𝑥𝐴𝑈 ∈ (𝐼𝑥)})
109fveq2d 6107 . . . 4 (𝑢 = 𝑈 → (#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) = (#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))
11 sneq 4135 . . . . . . 7 (𝑢 = 𝑈 → {𝑢} = {𝑈})
1211eqeq2d 2620 . . . . . 6 (𝑢 = 𝑈 → ((𝐼𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑈}))
1312rabbidv 3164 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})
1413fveq2d 6107 . . . 4 (𝑢 = 𝑈 → (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}) = (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}}))
1510, 14oveq12d 6567 . . 3 (𝑢 = 𝑈 → ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
16 eqid 2610 . . 3 (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
17 ovex 6577 . . 3 ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})) ∈ V
1815, 16, 17fvmpt 6191 . 2 (𝑈𝑉 → ((𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
197, 18eqtrd 2644 1 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {csn 4125   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   +𝑒 cxad 11820  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  VtxDegcvtxdg 40681 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-8 1979  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-pow 4769  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-vtxdg 40682 This theorem is referenced by:  vtxdgfival  40685  vtxdun  40696  vtxdlfgrval  40700  vtxd0nedgb  40703  vtxdushgrfvedg  40705
 Copyright terms: Public domain W3C validator