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Theorem vtxdgval 28714
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 28251 . . . 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 28713 . . . 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 6890 . 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 2821 . . . . . 6 (𝑒 = π‘ˆ β†’ (𝑒 ∈ (πΌβ€˜π‘₯) ↔ π‘ˆ ∈ (πΌβ€˜π‘₯)))
98rabbidv 3440 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)})
109fveq2d 6892 . . . 4 (𝑒 = π‘ˆ β†’ (β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}))
11 sneq 4637 . . . . . . 7 (𝑒 = π‘ˆ β†’ {𝑒} = {π‘ˆ})
1211eqeq2d 2743 . . . . . 6 (𝑒 = π‘ˆ β†’ ((πΌβ€˜π‘₯) = {𝑒} ↔ (πΌβ€˜π‘₯) = {π‘ˆ}))
1312rabbidv 3440 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}} = {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})
1413fveq2d 6892 . . . 4 (𝑒 = π‘ˆ β†’ (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}}))
1510, 14oveq12d 7423 . . 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 2732 . . 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 7438 . . 3 ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ π‘ˆ ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {π‘ˆ}})) ∈ V
1815, 16, 17fvmpt 6995 . 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 2772 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 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  {csn 4627   ↦ cmpt 5230  dom cdm 5675  β€˜cfv 6540  (class class class)co 7405   +𝑒 cxad 13086  β™―chash 14286  Vtxcvtx 28245  iEdgciedg 28246  VtxDegcvtxdg 28711
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-vtxdg 28712
This theorem is referenced by:  vtxdgfival  28715  vtxdun  28727  vtxdlfgrval  28731  vtxd0nedgb  28734  vtxdushgrfvedg  28736  vtxdginducedm1  28789
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