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Theorem vtxdgval 29501
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 29034 . . . 4 (𝑈𝑉𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdg‘𝐺)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 29500 . . . 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 6909 . 2 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈))
8 eleq1 2827 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∈ (𝐼𝑥) ↔ 𝑈 ∈ (𝐼𝑥)))
98rabbidv 3441 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴𝑢 ∈ (𝐼𝑥)} = {𝑥𝐴𝑈 ∈ (𝐼𝑥)})
109fveq2d 6911 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) = (♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))
11 sneq 4641 . . . . . . 7 (𝑢 = 𝑈 → {𝑢} = {𝑈})
1211eqeq2d 2746 . . . . . 6 (𝑢 = 𝑈 → ((𝐼𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑈}))
1312rabbidv 3441 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})
1413fveq2d 6911 . . . 4 (𝑢 = 𝑈 → (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}}))
1510, 14oveq12d 7449 . . 3 (𝑢 = 𝑈 → ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
16 eqid 2735 . . 3 (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
17 ovex 7464 . . 3 ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})) ∈ V
1815, 16, 17fvmpt 7016 . 2 (𝑈𝑉 → ((𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
197, 18eqtrd 2775 1 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  {csn 4631  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431   +𝑒 cxad 13150  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  VtxDegcvtxdg 29498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-vtxdg 29499
This theorem is referenced by:  vtxdgfival  29502  vtxdun  29514  vtxdlfgrval  29518  vtxd0nedgb  29521  vtxdushgrfvedg  29523  vtxdginducedm1  29576
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