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Theorem vtxdeqd 29496
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 5915 . . . . . 6 (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
42fveq1d 6907 . . . . . . 7 (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
54eleq2d 2826 . . . . . 6 (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
63, 5rabeqbidv 3454 . . . . 5 (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
76fveq2d 6909 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
84eqeq1d 2738 . . . . . 6 (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
93, 8rabeqbidv 3454 . . . . 5 (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
109fveq2d 6909 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
117, 10oveq12d 7450 . . 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 5232 . 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 2736 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
15 eqid 2736 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
16 eqid 2736 . . . 4 dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
1714, 15, 16vtxdgfval 29486 . . 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 2736 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
21 eqid 2736 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
22 eqid 2736 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
2320, 21, 22vtxdgfval 29486 . . 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 2786 1 (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435  {csn 4625  cmpt 5224  dom cdm 5684  cfv 6560  (class class class)co 7432   +𝑒 cxad 13153  chash 14370  Vtxcvtx 29014  iEdgciedg 29015  VtxDegcvtxdg 29484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-vtxdg 29485
This theorem is referenced by:  eupthvdres  30255
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