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Theorem vtxdgfval 40683
 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 40682 . . 3 VtxDeg = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
21a1i 11 . 2 (𝐺𝑊 → VtxDeg = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}})))))
3 fvex 6113 . . . 4 (Vtx‘𝑔) ∈ V
4 fvex 6113 . . . 4 (iEdg‘𝑔) ∈ V
5 simpl 472 . . . . 5 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
6 dmeq 5246 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
7 fveq1 6102 . . . . . . . . . 10 (𝑒 = (iEdg‘𝑔) → (𝑒𝑥) = ((iEdg‘𝑔)‘𝑥))
87eleq2d 2673 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
96, 8rabeqbidv 3168 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
109fveq2d 6107 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) = (#‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
117eqeq1d 2612 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → ((𝑒𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
126, 11rabeqbidv 3168 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
1312fveq2d 6107 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}) = (#‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
1410, 13oveq12d 6567 . . . . . 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}})))
1514adantl 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}})))
165, 15mpteq12dv 4663 . . . 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}}))))
173, 4, 16csbie2 3529 . . 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}})))
18 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
19 vtxdgfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2018, 19syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
21 fveq2 6103 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2221dmeqd 5248 . . . . . . . . 9 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom (iEdg‘𝐺))
23 vtxdgfval.a . . . . . . . . . 10 𝐴 = dom 𝐼
24 vtxdgfval.i . . . . . . . . . . 11 𝐼 = (iEdg‘𝐺)
2524dmeqi 5247 . . . . . . . . . 10 dom 𝐼 = dom (iEdg‘𝐺)
2623, 25eqtri 2632 . . . . . . . . 9 𝐴 = dom (iEdg‘𝐺)
2722, 26syl6eqr 2662 . . . . . . . 8 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
2821, 24syl6eqr 2662 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
2928fveq1d 6105 . . . . . . . . 9 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼𝑥))
3029eleq2d 2673 . . . . . . . 8 (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼𝑥)))
3127, 30rabeqbidv 3168 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥𝐴𝑢 ∈ (𝐼𝑥)})
3231fveq2d 6107 . . . . . 6 (𝑔 = 𝐺 → (#‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}))
3329eqeq1d 2612 . . . . . . . 8 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑢}))
3427, 33rabeqbidv 3168 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})
3534fveq2d 6107 . . . . . 6 (𝑔 = 𝐺 → (#‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))
3632, 35oveq12d 6567 . . . . 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}})))
3720, 36mpteq12dv 4663 . . . 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}}))))
3837adantl 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}}))))
3917, 38syl5eq 2656 . 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}}))))
40 elex 3185 . 2 (𝐺𝑊𝐺 ∈ V)
41 fvex 6113 . . . 4 (Vtx‘𝐺) ∈ V
4219, 41eqeltri 2684 . . 3 𝑉 ∈ V
43 mptexg 6389 . . 3 (𝑉 ∈ V → (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) ∈ V)
4442, 43mp1i 13 . 2 (𝐺𝑊 → (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))) ∈ V)
452, 39, 40, 44fvmptd 6197 1 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499  {csn 4125   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   +𝑒 cxad 11820  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  VtxDegcvtxdg 40681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-vtxdg 40682 This theorem is referenced by:  vtxdgval  40684  vtxdgf  40686  vtxdeqd  40692
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