vtxdfifiun - Intuitionistic Logic Explorer

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Theorem vtxdfifiun 16452

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 2537	. . . . . . . 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 4978	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐼 ∪ 𝐽))
4		dmun 4983	. . . . . . . . . . . . . 14 ⊢ dom (𝐼 ∪ 𝐽) = (dom 𝐼 ∪ dom 𝐽)
5	3, 4	eqtrdi 2287	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
6	5	eleq2d 2308	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7		elun 3370	. . . . . . . . . . . 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 469	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10		andir 831	. . . . . . . . . 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 2358	. . . . . . . 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 2283	. . . . . . 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 3498	. . . . . . . . 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 2242	. . . . . . . 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 2537	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
18	2	fveq1d 5692	. . . . . . . . . . . . 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 5403	. . . . . . . . . . . . . 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 5403	. . . . . . . . . . . . . 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 5763	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥))
30	19, 29	eqtrd 2271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼‘𝑥))
31	30	eleq2d 2308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑥)))
32	31	rabbidva 2809	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
33	17, 32	eqtr3id 2285	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
34		df-rab 2537	. . . . . . . . 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 5764	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥))
41	35, 40	eqtrd 2271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽‘𝑥))
42	41	eleq2d 2308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽‘𝑥)))
43	42	rabbidva 2809	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
44	34, 43	eqtr3id 2285	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
45	33, 44	uneq12d 3384	. . . . . . 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 2275	. . . . . 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 5694	. . . . 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 2238	. . . . . . 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 16444	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ Fin)
56		eqid 2238	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
57		vtxdun.j	. . . . . . 7 ⊢ 𝐽 = (iEdg‘𝐻)
58		eqid 2238	. . . . . . 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 2315	. . . . . . 7 ⊢ (𝜑 → (Vtx‘𝐻) ∈ Fin)
62	53, 60	eleqtrrd 2318	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐻))
63		vtxdfifiun.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
64	56, 57, 58, 59, 61, 62, 63	vtxedgfi 16444	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ Fin)
65		ssrab2 3333	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼
66		ssrab2 3333	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽
67		ss2in 3459	. . . . . . . . 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 430	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
69	68, 26	sseqtrid 3298	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅)
70		ss0 3563	. . . . . . 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 11223	. . . . . 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 1278	. . . . 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 2271	. . . 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 2537	. . . . . . . 8 ⊢ {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
76	8	anbi1d 469	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
77		andir 831	. . . . . . . . . 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 2358	. . . . . . . 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 2283	. . . . . . 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 3498	. . . . . . . . 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 2242	. . . . . . . 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 2537	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
85	30	eqeq1d 2247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼‘𝑥) = {𝑁}))
86	85	rabbidva 2809	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
87	84, 86	eqtr3id 2285	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
88		df-rab 2537	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
89	41	eqeq1d 2247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽‘𝑥) = {𝑁}))
90	89	rabbidva 2809	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
91	88, 90	eqtr3id 2285	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
92	87, 91	uneq12d 3384	. . . . . . 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 2275	. . . . . 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 5694	. . . . 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 16445	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ Fin)
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16445	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ Fin)
97		ssrab2 3333	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼
98		ssrab2 3333	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽
99		ss2in 3459	. . . . . . . . 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 430	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
101	100, 26	sseqtrid 3298	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅)
102		ss0 3563	. . . . . . 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 11223	. . . . . 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 1278	. . . . 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 2271	. . . 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 6093	. . 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 11198	. . . . . 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 9601	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℂ)
111		hashcl 11198	. . . . . 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 9601	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℂ)
114		hashcl 11198	. . . . . 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 9601	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℂ)
117		hashcl 11198	. . . . . 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 9601	. . . 4 ⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℂ)
120	110, 113, 116, 119	add4d 8485	. . 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 2271	. 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 2238	. . 3 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
123		eqid 2238	. . 3 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
124		eqid 2238	. . 3 ⊢ dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
125		unfidisj 7219	. . . . 5 ⊢ ((dom 𝐼 ∈ Fin ∧ dom 𝐽 ∈ Fin ∧ (dom 𝐼 ∩ dom 𝐽) = ∅) → (dom 𝐼 ∪ dom 𝐽) ∈ Fin)
126	51, 59, 26, 125	syl3anc 1278	. . . 4 ⊢ (𝜑 → (dom 𝐼 ∪ dom 𝐽) ∈ Fin)
127	5, 126	eqeltrd 2315	. . 3 ⊢ (𝜑 → dom (iEdg‘𝑈) ∈ Fin)
128		vtxdun.vu	. . . 4 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
129	128, 52	eqeltrd 2315	. . 3 ⊢ (𝜑 → (Vtx‘𝑈) ∈ Fin)
130	53, 128	eleqtrrd 2318	. . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝑈))
131	122	1vgrex 16175	. . . . 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 16281	. . 3 ⊢ (𝜑 → 𝑈 ∈ UPGraph)
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16446	. 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 16446	. . 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 16446	. . 3 ⊢ (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) + (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
137	135, 136	oveq12d 6093	. 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 2281	1 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 720 = wceq 1402 ∈ wcel 2209 {cab 2224 {crab 2532 Vcvv 2821 ∪ cun 3218 ∩ cin 3219 ⊆ wss 3220 ∅c0 3520 {csn 3705 dom cdm 4769 Fun wfun 5366 Fn wfn 5367 ‘cfv 5372 (class class class)co 6075 Fincfn 7012 + caddc 8172 ℕ₀cn0 9542 ♯chash 11192 Vtxcvtx 16167 iEdgciedg 16168 UPGraphcupgr 16246 VtxDegcvtxdg 16441

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-ltadd 8285

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-dec 9757 df-uz 9901 df-xadd 10154 df-ihash 11193 df-ndx 13333 df-slot 13334 df-base 13336 df-edgf 16160 df-vtx 16169 df-iedg 16170 df-upgren 16248 df-vtxdg 16442

This theorem is referenced by: p1evtxdeqfilem 16466 trlsegvdegfi 16622

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