Theorem vtxdeqd 16153

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 4933	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐻) = dom (iEdg‘𝐺))
4	2	fveq1d 5641	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
5	4	eleq2d 2301	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ((iEdg‘𝐻)‘𝑥) ↔ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)))
6	3, 5	rabeqbidv 2797	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)})
7	6	fveq2d 5643	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ 𝑢 ∈ ((iEdg‘𝐻)‘𝑥)}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}))
8	4	eqeq1d 2240	. . . . . 6 ⊢ (𝜑 → (((iEdg‘𝐻)‘𝑥) = {𝑢} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑢}))
9	3, 8	rabeqbidv 2797	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})
10	9	fveq2d 5643	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))
11	7, 10	oveq12d 6036	. . 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 4171	. 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 2231	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
15		eqid 2231	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
16		eqid 2231	. . . 4 ⊢ dom (iEdg‘𝐻) = dom (iEdg‘𝐻)
17	14, 15, 16	vtxdgfval 16145	. . 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 14	. 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 2231	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
21		eqid 2231	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22		eqid 2231	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
23	20, 21, 22	vtxdgfval 16145	. . 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 14	. 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 2274	1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  {crab 2514  {csn 3669   ↦ cmpt 4150  dom cdm 4725  ‘cfv 5326  (class class class)co 6018   +_𝑒 cxad 10005  ♯chash 11038  Vtxcvtx 15869  iEdgciedg 15870  VtxDegcvtxdg 16143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872  df-vtxdg 16144

This theorem is referenced by:  eupthvdres  16332