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Theorem vtxdgfval 29485
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 = (Vtx‘𝐺)
vtxdgfval.i 𝐼 = (iEdg‘𝐺)
vtxdgfval.a 𝐴 = dom 𝐼
Assertion
Ref Expression
vtxdgfval (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
Distinct variable groups:   𝑥,𝑢   𝑥,𝐴   𝑢,𝐺,𝑥   𝑢,𝑉
Allowed substitution hints:   𝐴(𝑢)   𝐼(𝑥,𝑢)   𝑉(𝑥)   𝑊(𝑥,𝑢)

Proof of Theorem vtxdgfval
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vtxdg 29484 . 2 VtxDeg = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
2 fvex 6919 . . . 4 (Vtx‘𝑔) ∈ V
3 fvex 6919 . . . 4 (iEdg‘𝑔) ∈ V
4 simpl 482 . . . . 5 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
5 dmeq 5914 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
6 fveq1 6905 . . . . . . . . . 10 (𝑒 = (iEdg‘𝑔) → (𝑒𝑥) = ((iEdg‘𝑔)‘𝑥))
76eleq2d 2827 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
85, 7rabeqbidv 3455 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
98fveq2d 6910 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
106eqeq1d 2739 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → ((𝑒𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
115, 10rabeqbidv 3455 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
1211fveq2d 6910 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
139, 12oveq12d 7449 . . . . . 6 (𝑒 = (iEdg‘𝑔) → ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
1413adantl 481 . . . . 5 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
154, 14mpteq12dv 5233 . . . 4 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → (𝑢𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))))
162, 3, 15csbie2 3938 . . 3 (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})))
17 fveq2 6906 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18 vtxdgfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1917, 18eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
20 fveq2 6906 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2120dmeqd 5916 . . . . . . . . 9 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom (iEdg‘𝐺))
22 vtxdgfval.a . . . . . . . . . 10 𝐴 = dom 𝐼
23 vtxdgfval.i . . . . . . . . . . 11 𝐼 = (iEdg‘𝐺)
2423dmeqi 5915 . . . . . . . . . 10 dom 𝐼 = dom (iEdg‘𝐺)
2522, 24eqtri 2765 . . . . . . . . 9 𝐴 = dom (iEdg‘𝐺)
2621, 25eqtr4di 2795 . . . . . . . 8 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
2720, 23eqtr4di 2795 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
2827fveq1d 6908 . . . . . . . . 9 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼𝑥))
2928eleq2d 2827 . . . . . . . 8 (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼𝑥)))
3026, 29rabeqbidv 3455 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥𝐴𝑢 ∈ (𝐼𝑥)})
3130fveq2d 6910 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}))
3228eqeq1d 2739 . . . . . . . 8 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑢}))
3326, 32rabeqbidv 3455 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})
3433fveq2d 6910 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))
3531, 34oveq12d 7449 . . . . 5 (𝑔 = 𝐺 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})) = ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})))
3619, 35mpteq12dv 5233 . . . 4 (𝑔 = 𝐺 → (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
3736adantl 481 . . 3 ((𝐺𝑊𝑔 = 𝐺) → (𝑢 ∈ (Vtx‘𝑔) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
3816, 37eqtrid 2789 . 2 ((𝐺𝑊𝑔 = 𝐺) → (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
39 elex 3501 . 2 (𝐺𝑊𝐺 ∈ V)
4018fvexi 6920 . . 3 𝑉 ∈ V
41 mptexg 7241 . . 3 (𝑉 ∈ V → (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) ∈ V)
4240, 41mp1i 13 . 2 (𝐺𝑊 → (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) ∈ V)
431, 38, 39, 42fvmptd2 7024 1 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  csb 3899  {csn 4626  cmpt 5225  dom cdm 5685  cfv 6561  (class class class)co 7431   +𝑒 cxad 13152  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  VtxDegcvtxdg 29483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-vtxdg 29484
This theorem is referenced by:  vtxdgval  29486  vtxdgop  29488  vtxdgf  29489  vtxdeqd  29495
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