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Theorem vtxdgval 26268
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 25799 . . . 4 (𝑈𝑉𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdg‘𝐺)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 26267 . . . 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 6155 . 2 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈))
8 eleq1 2686 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∈ (𝐼𝑥) ↔ 𝑈 ∈ (𝐼𝑥)))
98rabbidv 3180 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴𝑢 ∈ (𝐼𝑥)} = {𝑥𝐴𝑈 ∈ (𝐼𝑥)})
109fveq2d 6157 . . . 4 (𝑢 = 𝑈 → (#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) = (#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))
11 sneq 4163 . . . . . . 7 (𝑢 = 𝑈 → {𝑢} = {𝑈})
1211eqeq2d 2631 . . . . . 6 (𝑢 = 𝑈 → ((𝐼𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑈}))
1312rabbidv 3180 . . . . 5 (𝑢 = 𝑈 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})
1413fveq2d 6157 . . . 4 (𝑢 = 𝑈 → (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}) = (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}}))
1510, 14oveq12d 6628 . . 3 (𝑢 = 𝑈 → ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
16 eqid 2621 . . 3 (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
17 ovex 6638 . . 3 ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})) ∈ V
1815, 16, 17fvmpt 6244 . 2 (𝑈𝑉 → ((𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
197, 18eqtrd 2655 1 (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  {csn 4153  cmpt 4678  dom cdm 5079  cfv 5852  (class class class)co 6610   +𝑒 cxad 11896  #chash 13065  Vtxcvtx 25791  iEdgciedg 25792  VtxDegcvtxdg 26265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-vtxdg 26266
This theorem is referenced by:  vtxdgfival  26269  vtxdun  26280  vtxdlfgrval  26284  vtxd0nedgb  26287  vtxdushgrfvedg  26289
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