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Theorem vdgrfval 26422
 Description: The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
vdgrfval ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))))
Distinct variable groups:   𝑥,𝑢,𝐴   𝑢,𝐸,𝑥   𝑢,𝑉,𝑥   𝑢,𝑊,𝑥   𝑢,𝑋,𝑥

Proof of Theorem vdgrfval
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vdgr 26421 . . 3 VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
21a1i 11 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}})))))
3 simprl 790 . . 3 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → 𝑣 = 𝑉)
4 simprr 792 . . . . . . . 8 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → 𝑒 = 𝐸)
54dmeqd 5248 . . . . . . 7 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → dom 𝑒 = dom 𝐸)
6 simpl2 1058 . . . . . . . 8 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → 𝐸 Fn 𝐴)
7 fndm 5904 . . . . . . . 8 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
86, 7syl 17 . . . . . . 7 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → dom 𝐸 = 𝐴)
95, 8eqtrd 2644 . . . . . 6 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → dom 𝑒 = 𝐴)
104fveq1d 6105 . . . . . . 7 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑒𝑥) = (𝐸𝑥))
1110eleq2d 2673 . . . . . 6 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑢 ∈ (𝑒𝑥) ↔ 𝑢 ∈ (𝐸𝑥)))
129, 11rabeqbidv 3168 . . . . 5 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)} = {𝑥𝐴𝑢 ∈ (𝐸𝑥)})
1312fveq2d 6107 . . . 4 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) = (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}))
1410eqeq1d 2612 . . . . . 6 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑒𝑥) = {𝑢} ↔ (𝐸𝑥) = {𝑢}))
159, 14rabeqbidv 3168 . . . . 5 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})
1615fveq2d 6107 . . . 4 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}) = (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))
1713, 16oveq12d 6567 . . 3 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}})) = ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})))
183, 17mpteq12dv 4663 . 2 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))))
19 elex 3185 . . 3 (𝑉𝑊𝑉 ∈ V)
20193ad2ant1 1075 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → 𝑉 ∈ V)
21 fnex 6386 . . 3 ((𝐸 Fn 𝐴𝐴𝑋) → 𝐸 ∈ V)
22213adant1 1072 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → 𝐸 ∈ V)
23 mptexg 6389 . . 3 (𝑉𝑊 → (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))) ∈ V)
24233ad2ant1 1075 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))) ∈ V)
252, 18, 20, 22, 24ovmpt2d 6686 1 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {csn 4125   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   +𝑒 cxad 11820  #chash 12979   VDeg cvdg 26420 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-oprab 6553  df-mpt2 6554  df-vdgr 26421 This theorem is referenced by:  vdgrval  26423  vdgrf  26425  vdgrfif  26426
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