MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtxdgfval Structured version   Visualization version   GIF version

Theorem vtxdgfval 27255
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 27254 . 2 VtxDeg = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
2 fvex 6665 . . . 4 (Vtx‘𝑔) ∈ V
3 fvex 6665 . . . 4 (iEdg‘𝑔) ∈ V
4 simpl 486 . . . . 5 ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (iEdg‘𝑔)) → 𝑣 = (Vtx‘𝑔))
5 dmeq 5749 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → dom 𝑒 = dom (iEdg‘𝑔))
6 fveq1 6651 . . . . . . . . . 10 (𝑒 = (iEdg‘𝑔) → (𝑒𝑥) = ((iEdg‘𝑔)‘𝑥))
76eleq2d 2899 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → (𝑢 ∈ (𝑒𝑥) ↔ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)))
85, 7rabeqbidv 3461 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)})
98fveq2d 6656 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}))
106eqeq1d 2824 . . . . . . . . 9 (𝑒 = (iEdg‘𝑔) → ((𝑒𝑥) = {𝑢} ↔ ((iEdg‘𝑔)‘𝑥) = {𝑢}))
115, 10rabeqbidv 3461 . . . . . . . 8 (𝑒 = (iEdg‘𝑔) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}})
1211fveq2d 6656 . . . . . . 7 (𝑒 = (iEdg‘𝑔) → (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}))
139, 12oveq12d 7158 . . . . . 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 485 . . . . 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 5127 . . . 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 3894 . . 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 6652 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18 vtxdgfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1917, 18eqtr4di 2875 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
20 fveq2 6652 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2120dmeqd 5751 . . . . . . . . 9 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom (iEdg‘𝐺))
22 vtxdgfval.a . . . . . . . . . 10 𝐴 = dom 𝐼
23 vtxdgfval.i . . . . . . . . . . 11 𝐼 = (iEdg‘𝐺)
2423dmeqi 5750 . . . . . . . . . 10 dom 𝐼 = dom (iEdg‘𝐺)
2522, 24eqtri 2845 . . . . . . . . 9 𝐴 = dom (iEdg‘𝐺)
2621, 25eqtr4di 2875 . . . . . . . 8 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = 𝐴)
2720, 23eqtr4di 2875 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
2827fveq1d 6654 . . . . . . . . 9 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘𝑥) = (𝐼𝑥))
2928eleq2d 2899 . . . . . . . 8 (𝑔 = 𝐺 → (𝑢 ∈ ((iEdg‘𝑔)‘𝑥) ↔ 𝑢 ∈ (𝐼𝑥)))
3026, 29rabeqbidv 3461 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)} = {𝑥𝐴𝑢 ∈ (𝐼𝑥)})
3130fveq2d 6656 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ 𝑢 ∈ ((iEdg‘𝑔)‘𝑥)}) = (♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}))
3228eqeq1d 2824 . . . . . . . 8 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘𝑥) = {𝑢} ↔ (𝐼𝑥) = {𝑢}))
3326, 32rabeqbidv 3461 . . . . . . 7 (𝑔 = 𝐺 → {𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}})
3433fveq2d 6656 . . . . . 6 (𝑔 = 𝐺 → (♯‘{𝑥 ∈ dom (iEdg‘𝑔) ∣ ((iEdg‘𝑔)‘𝑥) = {𝑢}}) = (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))
3531, 34oveq12d 7158 . . . . 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 5127 . . . 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 485 . . 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, 37syl5eq 2869 . 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 3487 . 2 (𝐺𝑊𝐺 ∈ V)
4018fvexi 6666 . . 3 𝑉 ∈ V
41 mptexg 6966 . . 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 6758 1 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((♯‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134  Vcvv 3469  csb 3855  {csn 4539  cmpt 5122  dom cdm 5532  cfv 6334  (class class class)co 7140   +𝑒 cxad 12493  chash 13686  Vtxcvtx 26787  iEdgciedg 26788  VtxDegcvtxdg 27253
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-vtxdg 27254
This theorem is referenced by:  vtxdgval  27256  vtxdgop  27258  vtxdgf  27259  vtxdeqd  27265
  Copyright terms: Public domain W3C validator