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Theorem vtxdgfval 28464
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 28463 . 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 6859 . . . 4 (Vtxβ€˜π‘”) ∈ V
3 fvex 6859 . . . 4 (iEdgβ€˜π‘”) ∈ V
4 simpl 484 . . . . 5 ((𝑣 = (Vtxβ€˜π‘”) ∧ 𝑒 = (iEdgβ€˜π‘”)) β†’ 𝑣 = (Vtxβ€˜π‘”))
5 dmeq 5863 . . . . . . . . 9 (𝑒 = (iEdgβ€˜π‘”) β†’ dom 𝑒 = dom (iEdgβ€˜π‘”))
6 fveq1 6845 . . . . . . . . . 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 3423 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ 𝑒 ∈ (π‘’β€˜π‘₯)} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)})
98fveq2d 6850 . . . . . . 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 3423 . . . . . . . 8 (𝑒 = (iEdgβ€˜π‘”) β†’ {π‘₯ ∈ dom 𝑒 ∣ (π‘’β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}})
1211fveq2d 6850 . . . . . . 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 7379 . . . . . 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 5200 . . . 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 3901 . . 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 6846 . . . . . 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 6846 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
2120dmeqd 5865 . . . . . . . . 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 5864 . . . . . . . . . 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 6848 . . . . . . . . 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 3423 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ 𝑒 ∈ ((iEdgβ€˜π‘”)β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑒 ∈ (πΌβ€˜π‘₯)})
3130fveq2d 6850 . . . . . 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 3423 . . . . . . 7 (𝑔 = 𝐺 β†’ {π‘₯ ∈ dom (iEdgβ€˜π‘”) ∣ ((iEdgβ€˜π‘”)β€˜π‘₯) = {𝑒}} = {π‘₯ ∈ 𝐴 ∣ (πΌβ€˜π‘₯) = {𝑒}})
3433fveq2d 6850 . . . . . 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 7379 . . . . 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 5200 . . . 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 3465 . 2 (𝐺 ∈ π‘Š β†’ 𝐺 ∈ V)
4018fvexi 6860 . . 3 𝑉 ∈ V
41 mptexg 7175 . . 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 6960 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 3406  Vcvv 3447  β¦‹csb 3859  {csn 4590   ↦ cmpt 5192  dom cdm 5637  β€˜cfv 6500  (class class class)co 7361   +𝑒 cxad 13039  β™―chash 14239  Vtxcvtx 27996  iEdgciedg 27997  VtxDegcvtxdg 28462
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-vtxdg 28463
This theorem is referenced by:  vtxdgval  28465  vtxdgop  28467  vtxdgf  28468  vtxdeqd  28474
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