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

Theorem vtxdgval 27261
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 26798 . . . 4 (𝑈𝑉𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdg‘𝐺)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 27260 . . . 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 6651 . 2 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈))
8 eleq1 2880 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∈ (𝐼𝑥) ↔ 𝑈 ∈ (𝐼𝑥)))
98rabbidv 3430 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴𝑢 ∈ (𝐼𝑥)} = {𝑥𝐴𝑈 ∈ (𝐼𝑥)})
109fveq2d 6653 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) = (♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))
11 sneq 4538 . . . . . . 7 (𝑢 = 𝑈 → {𝑢} = {𝑈})
1211eqeq2d 2812 . . . . . 6 (𝑢 = 𝑈 → ((𝐼𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑈}))
1312rabbidv 3430 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})
1413fveq2d 6653 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}}))
1510, 14oveq12d 7157 . . 3 (𝑢 = 𝑈 → ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
16 eqid 2801 . . 3 (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
17 ovex 7172 . . 3 ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})) ∈ V
1815, 16, 17fvmpt 6749 . 2 (𝑈𝑉 → ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
197, 18eqtrd 2836 1 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  {csn 4528  cmpt 5113  dom cdm 5523  cfv 6328  (class class class)co 7139   +𝑒 cxad 12497  chash 13690  Vtxcvtx 26792  iEdgciedg 26793  VtxDegcvtxdg 27258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-vtxdg 27259
This theorem is referenced by:  vtxdgfival  27262  vtxdun  27274  vtxdlfgrval  27278  vtxd0nedgb  27281  vtxdushgrfvedg  27283  vtxdginducedm1  27336
  Copyright terms: Public domain W3C validator