Theorem vtxdeqd 29513

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	⊢ (𝜑 → 𝐺 ∈ 𝑋)
vtxdeqd.h	⊢ (𝜑 → 𝐻 ∈ 𝑌)
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.

Step	Hyp	Ref	Expression
1		vtxdeqd.v	. . 3 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))
2		vtxdeqd.i	. . . . . . 7 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
3	2	dmeqd 5930	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
4	2	fveq1d 6922	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
5	4	eleq2d 2830	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
6	3, 5	rabeqbidv 3462	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
7	6	fveq2d 6924	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
8	4	eqeq1d 2742	. . . . . 6 ⊢ (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
9	3, 8	rabeqbidv 3462	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
10	9	fveq2d 6924	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
11	7, 10	oveq12d 7466	. . 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}})))
12	1, 11	mpteq12dv 5257	. 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 ⊢ (𝜑 → 𝐻 ∈ 𝑌)
14		eqid 2740	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
15		eqid 2740	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
16		eqid 2740	. . . 4 ⊢ dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
17	14, 15, 16	vtxdgfval 29503	. . 3 ⊢ (𝐻 ∈ 𝑌 → (VtxDeg‘𝐻) = (𝑢 ∈ (Vtx‘𝐻) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}))))
18	13, 17	syl 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 ⊢ (𝜑 → 𝐺 ∈ 𝑋)
20		eqid 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
21		eqid 2740	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22		eqid 2740	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
23	20, 21, 22	vtxdgfval 29503	. . 3 ⊢ (𝐺 ∈ 𝑋 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
24	19, 23	syl 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}}))))
25	12, 18, 24	3eqtr4d 2790	1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443  {csn 4648   ↦ cmpt 5249  dom cdm 5700  ‘cfv 6573  (class class class)co 7448   +_𝑒 cxad 13173  ♯chash 14379  Vtxcvtx 29031  iEdgciedg 29032  VtxDegcvtxdg 29501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-vtxdg 29502

This theorem is referenced by:  eupthvdres  30267