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Theorem vtxdgfval 28724
Description: The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
Hypotheses
Ref Expression
vtxdgfval.v 𝑉 = (Vtxβ€˜πΊ)
vtxdgfval.i 𝐼 = (iEdgβ€˜πΊ)
vtxdgfval.a 𝐴 = dom 𝐼
Assertion
Ref Expression
vtxdgfval (𝐺 ∈ π‘Š β†’ (VtxDegβ€˜πΊ) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
Distinct variable groups:   π‘₯,𝑒   π‘₯,𝐴   𝑒,𝐺,π‘₯   𝑒,𝑉
Allowed substitution hints:   𝐴(𝑒)   𝐼(π‘₯,𝑒)   𝑉(π‘₯)   π‘Š(π‘₯,𝑒)

Proof of Theorem vtxdgfval
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vtxdg 28723 . 2 VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtxβ€˜π‘”) / π‘£β¦Œβ¦‹(iEdgβ€˜π‘”) / π‘’β¦Œ(𝑒 ∈ 𝑣 ↦ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}}))))
2 fvex 6905 . . . 4 (Vtxβ€˜π‘”) ∈ V
3 fvex 6905 . . . 4 (iEdgβ€˜π‘”) ∈ V
4 simpl 484 . . . . 5 ((𝑣 = (Vtxβ€˜π‘”) ∧ 𝑒 = (iEdgβ€˜π‘”)) β†’ 𝑣 = (Vtxβ€˜π‘”))
5 dmeq 5904 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ dom 𝑒 = dom (iEdgβ€˜π‘”))
6 fveq1 6891 . . . . . . . . . 10 (𝑒 = (iEdgβ€˜π‘”) β†’ (π‘’β€˜π‘₯) = ((iEdgβ€˜π‘”)β€˜π‘₯))
76eleq2d 2820 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ (𝑒 ∈ (π‘’β€˜π‘₯) ↔ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)))
85, 7rabeqbidv 3450 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)})
98fveq2d 6896 . . . . . . 7 (𝑒 = (iEdgβ€˜π‘”) β†’ (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) = (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}))
106eqeq1d 2735 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ ((π‘’β€˜π‘₯) = {𝑒} ↔ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}))
115, 10rabeqbidv 3450 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})
1211fveq2d 6896 . . . . . . 7 (𝑒 = (iEdgβ€˜π‘”) β†’ (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}}) = (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}}))
139, 12oveq12d 7427 . . . . . 6 (𝑒 = (iEdgβ€˜π‘”) β†’ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}})) = ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})))
1413adantl 483 . . . . 5 ((𝑣 = (Vtxβ€˜π‘”) ∧ 𝑒 = (iEdgβ€˜π‘”)) β†’ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}})) = ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})))
154, 14mpteq12dv 5240 . . . 4 ((𝑣 = (Vtxβ€˜π‘”) ∧ 𝑒 = (iEdgβ€˜π‘”)) β†’ (𝑒 ∈ 𝑣 ↦ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ (Vtxβ€˜π‘”) ↦ ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}}))))
162, 3, 15csbie2 3936 . . 3 ⦋(Vtxβ€˜π‘”) / π‘£β¦Œβ¦‹(iEdgβ€˜π‘”) / π‘’β¦Œ(𝑒 ∈ 𝑣 ↦ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ (Vtxβ€˜π‘”) ↦ ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})))
17 fveq2 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
18 vtxdgfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
1917, 18eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
20 fveq2 6892 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2120dmeqd 5906 . . . . . . . . 9 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom (iEdgβ€˜πΊ))
22 vtxdgfval.a . . . . . . . . . 10 𝐴 = dom 𝐼
23 vtxdgfval.i . . . . . . . . . . 11 𝐼 = (iEdgβ€˜πΊ)
2423dmeqi 5905 . . . . . . . . . 10 dom 𝐼 = dom (iEdgβ€˜πΊ)
2522, 24eqtri 2761 . . . . . . . . 9 𝐴 = dom (iEdgβ€˜πΊ)
2621, 25eqtr4di 2791 . . . . . . . 8 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = 𝐴)
2720, 23eqtr4di 2791 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
2827fveq1d 6894 . . . . . . . . 9 (𝑔 = 𝐺 β†’ ((iEdgβ€˜π‘”)β€˜π‘₯) = (πΌβ€˜π‘₯))
2928eleq2d 2820 . . . . . . . 8 (𝑔 = 𝐺 β†’ (𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯) ↔ 𝑒 ∈ (πΌβ€˜π‘₯)))
3026, 29rabeqbidv 3450 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)})
3130fveq2d 6896 . . . . . 6 (𝑔 = 𝐺 β†’ (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}))
3228eqeq1d 2735 . . . . . . . 8 (𝑔 = 𝐺 β†’ (((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒} ↔ (πΌβ€˜π‘₯) = {𝑒}))
3326, 32rabeqbidv 3450 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})
3433fveq2d 6896 . . . . . 6 (𝑔 = 𝐺 β†’ (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))
3531, 34oveq12d 7427 . . . . 5 (𝑔 = 𝐺 β†’ ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})) = ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})))
3619, 35mpteq12dv 5240 . . . 4 (𝑔 = 𝐺 β†’ (𝑒 ∈ (Vtxβ€˜π‘”) ↦ ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
3736adantl 483 . . 3 ((𝐺 ∈ π‘Š ∧ 𝑔 = 𝐺) β†’ (𝑒 ∈ (Vtxβ€˜π‘”) ↦ ((β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
3816, 37eqtrid 2785 . 2 ((𝐺 ∈ π‘Š ∧ 𝑔 = 𝐺) β†’ ⦋(Vtxβ€˜π‘”) / π‘£β¦Œβ¦‹(iEdgβ€˜π‘”) / π‘’β¦Œ(𝑒 ∈ 𝑣 ↦ ((β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}}))) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
39 elex 3493 . 2 (𝐺 ∈ π‘Š β†’ 𝐺 ∈ V)
4018fvexi 6906 . . 3 𝑉 ∈ V
41 mptexg 7223 . . 3 (𝑉 ∈ V β†’ (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))) ∈ V)
4240, 41mp1i 13 . 2 (𝐺 ∈ π‘Š β†’ (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))) ∈ V)
431, 38, 39, 42fvmptd2 7007 1 (𝐺 ∈ π‘Š β†’ (VtxDegβ€˜πΊ) = (𝑒 ∈ 𝑉 ↦ ((β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}) +𝑒 (β™―β€˜{π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475  β¦‹csb 3894  {csn 4629   ↦ cmpt 5232  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409   +𝑒 cxad 13090  β™―chash 14290  Vtxcvtx 28256  iEdgciedg 28257  VtxDegcvtxdg 28722
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-vtxdg 28723
This theorem is referenced by:  vtxdgval  28725  vtxdgop  28727  vtxdgf  28728  vtxdeqd  28734
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