vtxdfifiun - Intuitionistic Logic Explorer

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Theorem vtxdfifiun 16292

Description: The degree of a vertex in the union of two pseudographs of finite size on the same finite vertex set is the sum of the degrees of the vertex in each pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)

**Hypotheses**
Ref	Expression
vtxdun.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxdun.j	⊢ 𝐽 = (iEdg‘𝐻)
vtxdun.vg	⊢ 𝑉 = (Vtx‘𝐺)
vtxdun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
vtxdun.vu	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
vtxdfifiun.v	⊢ (𝜑 → 𝑉 ∈ Fin)
vtxdfifiun.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
vtxdfifiun.h	⊢ (𝜑 → 𝐻 ∈ UPGraph)
vtxdun.d	⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
vtxdun.fi	⊢ (𝜑 → Fun 𝐼)
vtxdun.fj	⊢ (𝜑 → Fun 𝐽)
vtxdun.n	⊢ (𝜑 → 𝑁 ∈ 𝑉)
vtxdun.u	⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))
vtxdfiun.a	⊢ (𝜑 → dom 𝐼 ∈ Fin)
vtxdfiun.b	⊢ (𝜑 → dom 𝐽 ∈ Fin)

**Assertion**
Ref	Expression
vtxdfifiun	⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))

Proof of Theorem vtxdfifiun Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2529	. . . . . . . 8 ⊢ {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
2		vtxdun.u	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))
3	2	dmeqd 4958	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐼 ∪ 𝐽))
4		dmun 4963	. . . . . . . . . . . . . 14 ⊢ dom (𝐼 ∪ 𝐽) = (dom 𝐼 ∪ dom 𝐽)
5	3, 4	eqtrdi 2281	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
6	5	eleq2d 2302	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7		elun 3360	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽))
8	6, 7	bitrdi 196	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽)))
9	8	anbi1d 465	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10		andir 827	. . . . . . . . . 10 ⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
11	9, 10	bitrdi 196	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
12	11	abbidv 2352	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
13	1, 12	eqtrid 2277	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
14		unab 3488	. . . . . . . . 9 ⊢ ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))}
15	14	eqcomi 2236	. . . . . . . 8 ⊢ {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))})
16	15	a1i 9	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}))
17		df-rab 2529	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
18	2	fveq1d 5672	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
19	18	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
20		vtxdun.fi	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐼)
21	20	funfnd 5383	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
22	21	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
23		vtxdun.fj	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐽)
24	23	funfnd 5383	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 Fn dom 𝐽)
25	24	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
26		vtxdun.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
27	26	anim1i 340	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
28		fvun1 5743	. . . . . . . . . . . . 13 ⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥))
29	22, 25, 27, 28	syl3anc 1274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥))
30	19, 29	eqtrd 2265	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼‘𝑥))
31	30	eleq2d 2302	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑥)))
32	31	rabbidva 2801	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
33	17, 32	eqtr3id 2279	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
34		df-rab 2529	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
35	18	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
36	21	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
37	24	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
38	26	anim1i 340	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
39		fvun2 5744	. . . . . . . . . . . . 13 ⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥))
40	36, 37, 38, 39	syl3anc 1274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥))
41	35, 40	eqtrd 2265	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽‘𝑥))
42	41	eleq2d 2302	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽‘𝑥)))
43	42	rabbidva 2801	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
44	34, 43	eqtr3id 2279	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
45	33, 44	uneq12d 3374	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))
46	13, 16, 45	3eqtrd 2269	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))
47	46	fveq2d 5674	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
48		vtxdun.vg	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
49		vtxdun.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
50		eqid 2232	. . . . . . 7 ⊢ dom 𝐼 = dom 𝐼
51		vtxdfiun.a	. . . . . . 7 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
52		vtxdfifiun.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
53		vtxdun.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ 𝑉)
54		vtxdfifiun.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
55	48, 49, 50, 51, 52, 53, 54	vtxedgfi 16284	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ Fin)
56		eqid 2232	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
57		vtxdun.j	. . . . . . 7 ⊢ 𝐽 = (iEdg‘𝐻)
58		eqid 2232	. . . . . . 7 ⊢ dom 𝐽 = dom 𝐽
59		vtxdfiun.b	. . . . . . 7 ⊢ (𝜑 → dom 𝐽 ∈ Fin)
60		vtxdun.vh	. . . . . . . 8 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
61	60, 52	eqeltrd 2309	. . . . . . 7 ⊢ (𝜑 → (Vtx‘𝐻) ∈ Fin)
62	53, 60	eleqtrrd 2312	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐻))
63		vtxdfifiun.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
64	56, 57, 58, 59, 61, 62, 63	vtxedgfi 16284	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ Fin)
65		ssrab2 3323	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼
66		ssrab2 3323	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽
67		ss2in 3449	. . . . . . . . 9 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽))
68	65, 66, 67	mp2an 426	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
69	68, 26	sseqtrid 3288	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅)
70		ss0 3549	. . . . . . 7 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅)
71	69, 70	syl 14	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅)
72		hashun 11169	. . . . . 6 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ Fin ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ Fin ∧ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
73	55, 64, 71, 72	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
74	47, 73	eqtrd 2265	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
75		df-rab 2529	. . . . . . . 8 ⊢ {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
76	8	anbi1d 465	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
77		andir 827	. . . . . . . . . 10 ⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
78	76, 77	bitrdi 196	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))))
79	78	abbidv 2352	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
80	75, 79	eqtrid 2277	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
81		unab 3488	. . . . . . . . 9 ⊢ ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))}
82	81	eqcomi 2236	. . . . . . . 8 ⊢ {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})})
83	82	a1i 9	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}))
84		df-rab 2529	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
85	30	eqeq1d 2241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼‘𝑥) = {𝑁}))
86	85	rabbidva 2801	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
87	84, 86	eqtr3id 2279	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
88		df-rab 2529	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
89	41	eqeq1d 2241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽‘𝑥) = {𝑁}))
90	89	rabbidva 2801	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
91	88, 90	eqtr3id 2279	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
92	87, 91	uneq12d 3374	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))
93	80, 83, 92	3eqtrd 2269	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))
94	93	fveq2d 5674	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
95	48, 49, 50, 51, 52, 53, 54	vtxlpfi 16285	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ Fin)
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16285	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ Fin)
97		ssrab2 3323	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼
98		ssrab2 3323	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽
99		ss2in 3449	. . . . . . . . 9 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽))
100	97, 98, 99	mp2an 426	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
101	100, 26	sseqtrid 3288	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅)
102		ss0 3549	. . . . . . 7 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅)
103	101, 102	syl 14	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅)
104		hashun 11169	. . . . . 6 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ Fin ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ Fin ∧ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
105	95, 96, 103, 104	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
106	94, 105	eqtrd 2265	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
107	74, 106	oveq12d 6068	. . 3 ⊢ (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) + ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
108		hashcl 11144	. . . . . 6 ⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ Fin → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
109	55, 108	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℕ₀)
110	109	nn0cnd 9555	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℂ)
111		hashcl 11144	. . . . . 6 ⊢ ({𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ Fin → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℕ₀)
112	64, 111	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℕ₀)
113	112	nn0cnd 9555	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℂ)
114		hashcl 11144	. . . . . 6 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ Fin → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℕ₀)
115	95, 114	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℕ₀)
116	115	nn0cnd 9555	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℂ)
117		hashcl 11144	. . . . . 6 ⊢ ({𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ Fin → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℕ₀)
118	96, 117	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℕ₀)
119	118	nn0cnd 9555	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℂ)
120	110, 113, 116, 119	add4d 8442	. . 3 ⊢ (𝜑 → (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) + ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) + ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
121	107, 120	eqtrd 2265	. 2 ⊢ (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) + ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
122		eqid 2232	. . 3 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
123		eqid 2232	. . 3 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
124		eqid 2232	. . 3 ⊢ dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
125		unfidisj 7182	. . . . 5 ⊢ ((dom 𝐼 ∈ Fin ∧ dom 𝐽 ∈ Fin ∧ (dom 𝐼 ∩ dom 𝐽) = ∅) → (dom 𝐼 ∪ dom 𝐽) ∈ Fin)
126	51, 59, 26, 125	syl3anc 1274	. . . 4 ⊢ (𝜑 → (dom 𝐼 ∪ dom 𝐽) ∈ Fin)
127	5, 126	eqeltrd 2309	. . 3 ⊢ (𝜑 → dom (iEdg‘𝑈) ∈ Fin)
128		vtxdun.vu	. . . 4 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
129	128, 52	eqeltrd 2309	. . 3 ⊢ (𝜑 → (Vtx‘𝑈) ∈ Fin)
130	53, 128	eleqtrrd 2312	. . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝑈))
131	122	1vgrex 16015	. . . . 5 ⊢ (𝑁 ∈ (Vtx‘𝑈) → 𝑈 ∈ V)
132	130, 131	syl 14	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
133	54, 63, 49, 57, 48, 60, 26, 132, 128, 2	upgrun 16121	. . 3 ⊢ (𝜑 → 𝑈 ∈ UPGraph)
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16286	. 2 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
135	48, 49, 50, 51, 52, 53, 54	vtxdgfifival 16286	. . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})))
136	56, 57, 58, 59, 61, 62, 63	vtxdgfifival 16286	. . 3 ⊢ (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
137	135, 136	oveq12d 6068	. 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) + ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
138	121, 134, 137	3eqtr4d 2275	1 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 716 = wceq 1398 ∈ wcel 2203 {cab 2218 {crab 2524 Vcvv 2813 ∪ cun 3209 ∩ cin 3210 ⊆ wss 3211 ∅c0 3508 {csn 3689 dom cdm 4749 Fun wfun 5346 Fn wfn 5347 ‘cfv 5352 (class class class)co 6050 Fincfn 6975 + caddc 8130 ℕ₀cn0 9496 ♯chash 11138 Vtxcvtx 16007 iEdgciedg 16008 UPGraphcupgr 16086 VtxDegcvtxdg 16281

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-ltadd 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-2o 6648 df-oadd 6651 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-5 9299 df-6 9300 df-7 9301 df-8 9302 df-9 9303 df-n0 9497 df-z 9578 df-dec 9710 df-uz 9854 df-xadd 10106 df-ihash 11139 df-ndx 13215 df-slot 13216 df-base 13218 df-edgf 16000 df-vtx 16009 df-iedg 16010 df-upgren 16088 df-vtxdg 16282

This theorem is referenced by: p1evtxdeqfilem 16306 trlsegvdegfi 16462

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