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Theorem vtxdgval 28458
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 = (Vtxβ€˜πΊ)
vtxdgval.i 𝐼 = (iEdgβ€˜πΊ)
vtxdgval.a 𝐴 = dom 𝐼
Assertion
Ref Expression
vtxdgval (π‘ˆ ∈ 𝑉 β†’ ((VtxDegβ€˜πΊ)β€˜π‘ˆ) = ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐺   π‘₯,π‘ˆ
Allowed substitution hints:   𝐼(π‘₯)   𝑉(π‘₯)

Proof of Theorem vtxdgval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 vtxdgval.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
211vgrex 27995 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐺 ∈ V)
3 vtxdgval.i . . . . 5 𝐼 = (iEdgβ€˜πΊ)
4 vtxdgval.a . . . . 5 𝐴 = dom 𝐼
51, 3, 4vtxdgfval 28457 . . . 4 (𝐺 ∈ V β†’ (VtxDegβ€˜πΊ) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
62, 5syl 17 . . 3 (π‘ˆ ∈ 𝑉 β†’ (VtxDegβ€˜πΊ) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
76fveq1d 6849 . 2 (π‘ˆ ∈ 𝑉 β†’ ((VtxDegβ€˜πΊ)β€˜π‘ˆ) = ((𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})))β€˜π‘ˆ))
8 eleq1 2826 . . . . . 6 (𝑒 = π‘ˆ β†’ (𝑒 ∈ (πΌβ€˜π‘₯) ↔ π‘ˆ ∈ (πΌβ€˜π‘₯)))
98rabbidv 3418 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)})
109fveq2d 6851 . . . 4 (𝑒 = π‘ˆ β†’ (β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}))
11 sneq 4601 . . . . . . 7 (𝑒 = π‘ˆ β†’ {𝑒} = {π‘ˆ})
1211eqeq2d 2748 . . . . . 6 (𝑒 = π‘ˆ β†’ ((πΌβ€˜π‘₯) = {𝑒} ↔ (πΌβ€˜π‘₯) = {π‘ˆ}))
1312rabbidv 3418 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}} = {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})
1413fveq2d 6851 . . . 4 (𝑒 = π‘ˆ β†’ (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}}))
1510, 14oveq12d 7380 . . 3 (𝑒 = π‘ˆ β†’ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})) = ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})))
16 eqid 2737 . . 3 (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})))
17 ovex 7395 . . 3 ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})) ∈ V
1815, 16, 17fvmpt 6953 . 2 (π‘ˆ ∈ 𝑉 β†’ ((𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})))β€˜π‘ˆ) = ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})))
197, 18eqtrd 2777 1 (π‘ˆ ∈ 𝑉 β†’ ((VtxDegβ€˜πΊ)β€˜π‘ˆ) = ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3410  Vcvv 3448  {csn 4591   ↦ cmpt 5193  dom cdm 5638  β€˜cfv 6501  (class class class)co 7362   +𝑒 cxad 13038  β™―chash 14237  Vtxcvtx 27989  iEdgciedg 27990  VtxDegcvtxdg 28455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-vtxdg 28456
This theorem is referenced by:  vtxdgfival  28459  vtxdun  28471  vtxdlfgrval  28475  vtxd0nedgb  28478  vtxdushgrfvedg  28480  vtxdginducedm1  28533
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