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Theorem vtxdgfval 28762
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 28761 . 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 6904 . . . 4 (Vtxβ€˜π‘”) ∈ V
3 fvex 6904 . . . 4 (iEdgβ€˜π‘”) ∈ V
4 simpl 483 . . . . 5 ((𝑣 = (Vtxβ€˜π‘”) ∧ 𝑒 = (iEdgβ€˜π‘”)) β†’ 𝑣 = (Vtxβ€˜π‘”))
5 dmeq 5903 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ dom 𝑒 = dom (iEdgβ€˜π‘”))
6 fveq1 6890 . . . . . . . . . 10 (𝑒 = (iEdgβ€˜π‘”) β†’ (π‘’β€˜π‘₯) = ((iEdgβ€˜π‘”)β€˜π‘₯))
76eleq2d 2819 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ (𝑒 ∈ (π‘’β€˜π‘₯) ↔ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)))
85, 7rabeqbidv 3449 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)})
98fveq2d 6895 . . . . . . 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 2734 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ ((π‘’β€˜π‘₯) = {𝑒} ↔ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}))
115, 10rabeqbidv 3449 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})
1211fveq2d 6895 . . . . . . 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 7429 . . . . . 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 482 . . . . 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 5239 . . . 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 3935 . . 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 6891 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
18 vtxdgfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
1917, 18eqtr4di 2790 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
20 fveq2 6891 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2120dmeqd 5905 . . . . . . . . 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 5904 . . . . . . . . . 10 dom 𝐼 = dom (iEdgβ€˜πΊ)
2522, 24eqtri 2760 . . . . . . . . 9 𝐴 = dom (iEdgβ€˜πΊ)
2621, 25eqtr4di 2790 . . . . . . . 8 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = 𝐴)
2720, 23eqtr4di 2790 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
2827fveq1d 6893 . . . . . . . . 9 (𝑔 = 𝐺 β†’ ((iEdgβ€˜π‘”)β€˜π‘₯) = (πΌβ€˜π‘₯))
2928eleq2d 2819 . . . . . . . 8 (𝑔 = 𝐺 β†’ (𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯) ↔ 𝑒 ∈ (πΌβ€˜π‘₯)))
3026, 29rabeqbidv 3449 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)})
3130fveq2d 6895 . . . . . 6 (𝑔 = 𝐺 β†’ (β™―β€˜{π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)}) = (β™―β€˜{π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)}))
3228eqeq1d 2734 . . . . . . . 8 (𝑔 = 𝐺 β†’ (((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒} ↔ (πΌβ€˜π‘₯) = {𝑒}))
3326, 32rabeqbidv 3449 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})
3433fveq2d 6895 . . . . . 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 7429 . . . . 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 5239 . . . 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 482 . . 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 2784 . 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 3492 . 2 (𝐺 ∈ π‘Š β†’ 𝐺 ∈ V)
4018fvexi 6905 . . 3 𝑉 ∈ V
41 mptexg 7225 . . 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 7006 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 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  β¦‹csb 3893  {csn 4628   ↦ cmpt 5231  dom cdm 5676  β€˜cfv 6543  (class class class)co 7411   +𝑒 cxad 13092  β™―chash 14292  Vtxcvtx 28294  iEdgciedg 28295  VtxDegcvtxdg 28760
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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-vtxdg 28761
This theorem is referenced by:  vtxdgval  28763  vtxdgop  28765  vtxdgf  28766  vtxdeqd  28772
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