Theorem vtxdeqd 27267

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 5738	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
4	2	fveq1d 6647	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
5	4	eleq2d 2875	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
6	3, 5	rabeqbidv 3433	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
7	6	fveq2d 6649	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
8	4	eqeq1d 2800	. . . . . 6 ⊢ (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
9	3, 8	rabeqbidv 3433	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
10	9	fveq2d 6649	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
11	7, 10	oveq12d 7153	. . 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 5115	. 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 2798	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
15		eqid 2798	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
16		eqid 2798	. . . 4 ⊢ dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
17	14, 15, 16	vtxdgfval 27257	. . 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 2798	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
21		eqid 2798	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22		eqid 2798	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
23	20, 21, 22	vtxdgfval 27257	. . 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 2843	1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  {csn 4525   ↦ cmpt 5110  dom cdm 5519  ‘cfv 6324  (class class class)co 7135   +_𝑒 cxad 12493  ♯chash 13686  Vtxcvtx 26789  iEdgciedg 26790  VtxDegcvtxdg 27255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-vtxdg 27256

This theorem is referenced by:  eupthvdres  28020