MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtxdgval Structured version   Visualization version   GIF version

Theorem vtxdgval 29487
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 29020 . . . 4 (𝑈𝑉𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdg‘𝐺)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 29486 . . . 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 6907 . 2 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈))
8 eleq1 2828 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∈ (𝐼𝑥) ↔ 𝑈 ∈ (𝐼𝑥)))
98rabbidv 3443 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴𝑢 ∈ (𝐼𝑥)} = {𝑥𝐴𝑈 ∈ (𝐼𝑥)})
109fveq2d 6909 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) = (♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))
11 sneq 4635 . . . . . . 7 (𝑢 = 𝑈 → {𝑢} = {𝑈})
1211eqeq2d 2747 . . . . . 6 (𝑢 = 𝑈 → ((𝐼𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑈}))
1312rabbidv 3443 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})
1413fveq2d 6909 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}}))
1510, 14oveq12d 7450 . . 3 (𝑢 = 𝑈 → ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
16 eqid 2736 . . 3 (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
17 ovex 7465 . . 3 ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})) ∈ V
1815, 16, 17fvmpt 7015 . 2 (𝑈𝑉 → ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
197, 18eqtrd 2776 1 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  {csn 4625  cmpt 5224  dom cdm 5684  cfv 6560  (class class class)co 7432   +𝑒 cxad 13153  chash 14370  Vtxcvtx 29014  iEdgciedg 29015  VtxDegcvtxdg 29484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-vtxdg 29485
This theorem is referenced by:  vtxdgfival  29488  vtxdun  29500  vtxdlfgrval  29504  vtxd0nedgb  29507  vtxdushgrfvedg  29509  vtxdginducedm1  29562
  Copyright terms: Public domain W3C validator