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Theorem vtxdeqd 29510
Description: Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
Hypotheses
Ref Expression
vtxdeqd.g (𝜑𝐺𝑋)
vtxdeqd.h (𝜑𝐻𝑌)
vtxdeqd.v (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))
vtxdeqd.i (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
Assertion
Ref Expression
vtxdeqd (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Proof of Theorem vtxdeqd
Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vtxdeqd.v . . 3 (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))
2 vtxdeqd.i . . . . . . 7 (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
32dmeqd 5919 . . . . . 6 (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
42fveq1d 6909 . . . . . . 7 (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
54eleq2d 2825 . . . . . 6 (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
63, 5rabeqbidv 3452 . . . . 5 (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
76fveq2d 6911 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
84eqeq1d 2737 . . . . . 6 (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
93, 8rabeqbidv 3452 . . . . 5 (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
109fveq2d 6911 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
117, 10oveq12d 7449 . . 3 (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))
121, 11mpteq12dv 5239 . 2 (𝜑 → (𝑢 ∈ (Vtx‘𝐻) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}))) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
13 vtxdeqd.h . . 3 (𝜑𝐻𝑌)
14 eqid 2735 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
15 eqid 2735 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
16 eqid 2735 . . . 4 dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
1714, 15, 16vtxdgfval 29500 . . 3 (𝐻𝑌 → (VtxDeg‘𝐻) = (𝑢 ∈ (Vtx‘𝐻) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}))))
1813, 17syl 17 . 2 (𝜑 → (VtxDeg‘𝐻) = (𝑢 ∈ (Vtx‘𝐻) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}))))
19 vtxdeqd.g . . 3 (𝜑𝐺𝑋)
20 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
21 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
22 eqid 2735 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
2320, 21, 22vtxdgfval 29500 . . 3 (𝐺𝑋 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
2419, 23syl 17 . 2 (𝜑 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
2512, 18, 243eqtr4d 2785 1 (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  {csn 4631  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431   +𝑒 cxad 13150  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  VtxDegcvtxdg 29498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-vtxdg 29499
This theorem is referenced by:  eupthvdres  30264
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