Theorem vtxdeqd 29827

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 5895	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
4	2	fveq1d 6883	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
5	4	eleq2d 2849	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
6	3, 5	rabeqbidv 3434	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
7	6	fveq2d 6885	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
8	4	eqeq1d 2765	. . . . . 6 ⊢ (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
9	3, 8	rabeqbidv 3434	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
10	9	fveq2d 6885	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
11	7, 10	oveq12d 7428	. . 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 5198	. 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 2763	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
15		eqid 2763	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
16		eqid 2763	. . . 4 ⊢ dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
17	14, 15, 16	vtxdgfval 29817	. . 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 18	. 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 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
21		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22		eqid 2763	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
23	20, 21, 22	vtxdgfval 29817	. . 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 18	. 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 2808	1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {crab 3416  {csn 4589   ↦ cmpt 5192  dom cdm 5661  ‘cfv 6536  (class class class)co 7410   +_𝑒 cxad 13130  ♯chash 14362  Vtxcvtx 29346  iEdgciedg 29347  VtxDegcvtxdg 29815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-vtxdg 29816

This theorem is referenced by:  eupthvdres  30586