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Theorem vtxdgfval 26573
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 26572 . . 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 6362 . . . 4 (Vtx‘𝑔) ∈ V
4 fvex 6362 . . . 4 (iEdg‘𝑔) ∈ V
5 simpl 474 . . . . 5 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
6 dmeq 5479 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
7 fveq1 6351 . . . . . . . . . 10 (𝑒 = (iEdg‘𝑔) → (𝑒𝑥) = ((iEdg‘𝑔)‘𝑥))
87eleq2d 2825 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
96, 8rabeqbidv 3335 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
109fveq2d 6356 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
117eqeq1d 2762 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → ((𝑒𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
126, 11rabeqbidv 3335 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
1312fveq2d 6356 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
1410, 13oveq12d 6831 . . . . . 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 473 . . . . 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 4885 . . . 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 3704 . . 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 6352 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
19 vtxdgfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2018, 19syl6eqr 2812 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
21 fveq2 6352 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2221dmeqd 5481 . . . . . . . . 9 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom (iEdg‘𝐺))
23 vtxdgfval.a . . . . . . . . . 10 𝐴 = dom 𝐼
24 vtxdgfval.i . . . . . . . . . . 11 𝐼 = (iEdg‘𝐺)
2524dmeqi 5480 . . . . . . . . . 10 dom 𝐼 = dom (iEdg‘𝐺)
2623, 25eqtri 2782 . . . . . . . . 9 𝐴 = dom (iEdg‘𝐺)
2722, 26syl6eqr 2812 . . . . . . . 8 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
2821, 24syl6eqr 2812 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
2928fveq1d 6354 . . . . . . . . 9 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼𝑥))
3029eleq2d 2825 . . . . . . . 8 (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼𝑥)))
3127, 30rabeqbidv 3335 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥𝐴𝑢 ∈ (𝐼𝑥)})
3231fveq2d 6356 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}))
3329eqeq1d 2762 . . . . . . . 8 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑢}))
3427, 33rabeqbidv 3335 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})
3534fveq2d 6356 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))
3632, 35oveq12d 6831 . . . . 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 4885 . . . 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 473 . . 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 2806 . 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 3352 . 2 (𝐺𝑊𝐺 ∈ V)
41 fvex 6362 . . . 4 (Vtx‘𝐺) ∈ V
4219, 41eqeltri 2835 . . 3 𝑉 ∈ V
43 mptexg 6648 . . 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 6450 1 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  csb 3674  {csn 4321  cmpt 4881  dom cdm 5266  cfv 6049  (class class class)co 6813   +𝑒 cxad 12137  chash 13311  Vtxcvtx 26073  iEdgciedg 26074  VtxDegcvtxdg 26571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-vtxdg 26572
This theorem is referenced by:  vtxdgval  26574  vtxdgop  26576  vtxdgf  26577  vtxdeqd  26583
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